Complex Variables and the Laplace Transform for Engineers (Dover)~Tqw~_darksiderg

COMPLEX VARIABLES AND THE LAPLACE TRANSFORM FOR ENGINEERS COMPLEX VARIABLES AND TIIE LAPLACE TRANSFORM FOR ENGINEERS Wilbur R. LePage Departmentof ElectricalandComputerEngineering SyracuseUniversity Dover Publications, Inc. New York Copyright©1961byWilburR.LePage. AllrightsreservedunderPanAmericanand InternationalCopyrightConventions. Publishedin Canada by GeneralPublishing Com· pany,Ltd.,30LesmillRoad,DonMills,Toronto, Ontario. PublishedintheUnitedKingdombyConstable andCompany,Ltd.,10OrangeStreet,London WC2H7EG. ThisDoveredition,firstpublishedin1980,isan unabridgedandcorrectedrepublication ofthework originallypublishedin1961byMcGraw·Hill,Inc. InternationalStandardBookNumber:0·486-63926·6 Library of CongressCatalog CardNumber: 79-055908 Manufactured in theUnitedStates of America DoverPublications,Inc. 180 VariekStreet NewYork,N.Y.10014 ToTHOSEWHOfindsatisfactioninreflectivethought andwhoregardthescholarlyquestforunderstanding asinherentlyvaluabletotheindividualandtosociety, thisbookis dedicated PREFACE This book iswritten forthe serious student, probably at the graduate level,whoisinterestedinobtaininganunderstandingofthetheoryof FourierandLaplacetransforms,togetherwiththebasictheoryof functionsofacomplexvariable,withoutwhichthetransformtheory cannotbeunderstood.Noprior knowledgeother than agoodground- ingin the calculusisnecessary,althoughundoubtedly the material will havemoremeaningintheinitialstagesforthestudentwhobasthe motivationprovidedby someunderstanding ofthe simpler applications oftheLaplacetransform.Suchpriorknowledgewillusuallybeatan introductorylevel,havingtodowiththemechanicalmanipulationof formulas.Itisreasonabletobeginasubjectbythemanipulative approach,buttodososhouldleavetheseriousstudentinastateof unrest and perhaps mildconfusion.If he is alert,many ofthe manipu- lativeprocedureswillnotreallymakesense.If youhaveexperienced this kind of confusion and if it bothers you,you are ready to profit from astudy ofthis book,whichoccupiesapositionbetweenthe usual engi- neeringtreatments andthe abstracttreatmentsofthe mathematicians. Thebookieintendedtoprepareyouforcreativework,notmerelyto solvestereotypedproblems.Theapproachisintendedforworkersin anageofmaturetechnology,inwhichthescientificmethodoccupiesa position of dominance.Because of the heavy emphasis on interpretation and because of the lack of generality in the proofs, this should be regarded asan engineering book,in spite ofthe extensive use of mathematics. The highlypersonal aspect of the learning process makes it impossible foranauthortowriteabookthatisidealforanyoneexcepthimself. Recognition of this reality provides the key to howyou can benefit most fromabook such as this.Probably you will want firstto search for the main pattern of ideas, with the details to be filledin at such time as your interestisaroused.Learningisessentiallyarandomprocess,andan authorcannotinsistthat eventsin yourprogramoflearning willoccur in any predictable order.Therefore, it is recommended that you remain alerttopointsofinterest,andparticularlytopointsofconfusion.To acknowledge that a concept isnot fully understood isto recognize it as a pointofinterest.It issuggestedthatyougiveduerespectto sucha ix xPREFACE point,at the time it cries forattention,without regard forwhether it is the next topic in the book-searching for related ideas, referring to other texts, and, above all, experimenting with your own ideas.There is such awealth ofinterrelatednessoftopicsthat, if you dothis with complete intellectualhonesty,youwilleventuallyfindthatyouhavemorethan covered the text, without ever having read it in continuous fashion from cover to cover. Thetextisroughlyintwoparts.Thefirstpart,onfunctionsofa complexvariable,beginsatarelativelylowlevel.Experiencewith graduate students in electricalengineering at SyracuseUniversity,over aperiodoffiveyearsduringwhichthefirstelevenchaptersofthis material were used in note form, indicates that the approach is acceptable to mostbeginninggraduatestudents.Thelevelofdifficultygradually increasesthroughoutthebook,andthematerialbeyondChapter7 attains arelatively high degree ofsophistication.However,it isantici- patedthatwith agradualincreasein yourknowledgethe materialwill present an aspect of approximately constant difficulty. ThematerialonfunctionsofacomplexvariableiBquitesimilarto manyofthestandardbeginningengineering-orientedtextsonthesub- ject,exceptperhapsfortheamountofinterpretationandillustrative material.Oneotherdifferencewillbenoticedimmediately:theuseof 8=U+ jCJJinsteadoftheusualz=x+ iy.Tousejinplaceofi is established practice in engineering literature andprobably isnotcon- troversial.The choice of 8, u, and CJJin place of z, x, and y was a calculated riskintermsofreaderreaction.It providesaunityinthisonebook, but it willnecessitateasymboltranslationwhencomparingwithother books on function theory.My apologies are offered to anyone forwhom this is anuisance. Afewsuggestionsareofferedhere,tobothstudentandteacher,as towhatmaterialmightbeconsideredsuperfluousinaninitialcourse ofstudy.Chapter1providesmotivationandaperspectiveviewpoint. It willnotserveallstudentsequallywell,possiblybeingtooconcise forsomeandtooelementaryforothers.It hasnoessentialposition inthestreamofcontinuityandthereforecanbeomitted.Chapter2 gives the main introductory concepts and is essential.Chapter 3 should be coveredto the extent offirmlyestablishing the geometrical interpre- tationofafunctionofacomplexvariableasapointtransformation betweentwoplanes,andthebasicideasofconformalityofthetrans- formationasrelatedtoanalyticityofthefunction.However,onfirst readingitmaybeadvisablenottogointodetailsofalltheexamples given.Much of this is reference material. Chapters 4and 5are verybasicand should not be omitted.Chapter 6 bears a relation to the general text material similar to that of Chapter 3. PREFACExi Someknowledgeofmultivaluedfunctionsiscertainly essential,but the student shouldadjust to hisowntaste howmuchdetailandhowmany practicalillustrationsareappropriate.Muchofthischaptermaybe regardedasreferencematerial.Chapter7,thelastofthechapters devotedtofunctiontheory,consistsalmostcompletelyofreference materialpertinenttonetworktheory,andcanbeomittedwithoutloss ofcontinuity.Infacttheencyclopedicnatureofthischaptercauses some of the topics to appear out of logical order. Chapter8containsbackgroundoncertainpropertiesofintegrals, particularly improper integrals, in anticipation of applications in the later chapters.This chapter dealswith difficultmathematical concepts and, compared with the standards ofrigor set in the other chapters, is largely intuitive.The main purpose is to alert the reader to the major problems arisingwhenan improper integral isusedto representafunction.The chapter can be skipped without lossof continuity, but at least acursory reading is recommended, followed by deeper consideration of appropriate parts whilestudying the later chapters. Chapters9and10formthecoreofthesecondpartofthebook- theFourierandLaplacetransformtheory.In thetransitionfromthe Fourier integral to the one-sidedLaplace integral,the two-sidedLaplace integralisintroduced,onthe argumentthat conceptuallythe two-sided Laplaceintegralliesmidwaybetweentheothertwo.Thisseemsto smooth the way for the student to negotiate the subtle conceptual bridge betweentheFourierandLaplacetransformtheories.If theLaplace transformtheoryistobe understoodat the levelintended,there seems to benoalternative but to include the two-sidedLaplace transform. The theory of convolution integrals presented in Chapter 11is certainly fundamental, although not wholly a part of Laplace transform theory, and thereforeshouldbeincludedinacomprehensivecourseofstudy.The remainingchaptersdealwithspecialtopics,andeachhasitsroots in the all-important Chapter 10.Chapter 12 is essentially a continuation ofChapter10and isprimarily areference work.The practical applica- tions treated in Chapter 13 provide a brief summary of the theory of linear systemsandpreferablyshouldbestudiedtogetherwith,orfollowing,a course in network theory.Otherwise the treatment may be too abstract. However,takenatthepropertime,itcanbehelpfulinunifyingthe ideas about this important fieldofapplication. In regard to Chapter 14, on impulse functions,acritical response from somereadersisanticipatedbysayingthatthischapterrepresentsone particularviewpoint.Manywillsaythat 'it laborsthepointandthat allthe usefulideascontained therein can bereducedto onepage.This isamatterofopinion,anditisthoughtthatasignificantnumberof studentscanbenefitfromthisanalysis.Everyoneknowsthatimpulse PREFACE functions are not functions in the true sense of the word, and this chapter hassomethingtosayaboutthisquestion,castingtheusualresultsin suchaformthat nodoubtscanariseastothe meaning.Aknowledge of the customary formalisms associated with impulse functions and their symbolic transforms represents a bare minimum of accomplishment in this chapter. Finally, Chapters 15 and 16 on periodic functions and the Ztransform arerelatedandprovidebackgroundmaterialformanyofthepractical applications which you willprobably study elsewhere.The justification forincludingthesechaptersistobefoundinthedesiretopresentthis fundamentalappliedmaterialwiththesamedegreeofcompletenessas the Laplacetransform itself. Noparticular claimismade fororiginalityin thebasictheory,other than in organization and details ofpresentation.Nor isit claimed that theproofsarealwaysasshortoraselegantaspossible.Thegeneral criterionusedwasto selectproofsthat are realistically straightforward, with the hope that this would ensure a high degree of intellectual honesty, while always keeping intouch with simple concepts. Initspreliminaryversions,thismaterialhasbeentaughtbyabout twentydifferentcolleagues.Allofthesepersonshavemadehelp- fulsuggestions,andalistoftheirnameswouldbetoolongtogivein itsentirety.However,IwouldliketosingleoutProfessorsNorman Balabanian,DavidCheng,Harry Gruenberg,RichardMcFee,Fazlollah Reza,and Sundaram Seshu as having been especially helpful.Also,Pro- fessorRajendraNanavatiandMessrs.JosephCornacchioandRobert Richardsondeservespecialacknowledgmentforreadingandconstruc- tivelycriticizingtheentiremanuscript.Similaracknowledgmentis madetoProfessorsErikHemmingsenandJeromeBlackman,ofthe Syracuse Mathematics Department, for their careful reviews of Chapters 8through12.Finally,mysincerethanksgotoMissAnneL.Woods forherskillanduntiringeffortsintypingthevariousversionsofthe notesandthefinalmanuscript. WilburR.LePage CONTENTS Preface.ix Chapter1.Conceptual Structure ofSystem Analysis.1 1-1Introduction1 1-2ClassicalSteady-stateResponse of aLinear System1 1-3CharacterizationoftheSystemFunctionasaFunctionofaComplex Variable.2 1-4Fourier Series.5 1-5Fourier Integral6 1-6The LaplaceIntegral8 1-7Frequency, and theGeneralizedFrequency Variable10 1-8Stability.12 1-9Convolution-type Integrals12 1-10Idealized Systems.13 1-11Linear Systems withTime-varying Parameters14 1-12Other Systems.14 Problems14 :-Chapter 2.IntroductiontoFunction Theory19 2-1Introduction19 2-2Definition of aFunction24 2-3Limit,Continuity.26 2-4Derivative ofaFunction29 2-5Definition ofRegularity,Singular Points.andAnalyticity31 2-6The Cauchy-Riemann Equations.33 2-7TranscendentalFunctions.35 2-8HarmonicFunctions41 Problems42 ChapterI.ConformalMapping.46 3-1Introduction46 3-2SomeSimpleExamples ofTransformations.46 3:3Practical Applications.52 3-4The Function w=1/8.56 3-5The Function w=~ ( 3+ 1/8)57 3-6TheExponential Function61 3-7Hyperbolicand TrigonometricFunctions62 3-8The Point at Infinity; The Riemann Sphere64 3-9Further Properties of theReciprocalFunction66 3-10The Bilinear Transformation.70 3-11ConformalMapping.73 xiii xivCONTENTS 3-12Solution of Two-dimensional-fieldProblems Problems k Chapter ,.IntegratioD 4-1Introduction 4-2SomeDefinitions 4-3Integration. 4-4UpperBound of aContour Integral. 4-5Cauchy Integral Theorem. 4-6Independence of IntegrationPath. 4-7Significance of Connectivity. 4-8Primitive Function(Antiderivative) 4-9The Logarithm. 4-10Cauchy Integral Formulas 4-11Implications of the Cauchy Integral Formulas. 4-12Morera's Theorem. 4-13Use of Primitive Function to Evaluate aContour Integral Problems 77 81 SS SS SS 88 94 94 98 99 100 102 105 108 109 109 110 fChapter I.Infinite Series.116 5-1Introduction116 5-2Series of Constants.116 5-3Series of Functions.120 5-4Integration of Series124 5-5Convergence of Power Series.125 5-6Properties of Power Series128 5-7Taylor Series129 5-8Laurent Series.134 5-9Comparison of Taylor and Laurent Series136 5-10Laurent Expansions about aSingular Point139 5-11PolesandEssential Singularities;Residues.142 5-12Residue Theorem.145 5-13AnalyticContinuation.147 5-14Classification of Single-valuedFunctions152 5-15Partial-fractionExpansion153 5-16Partial-fractionExpansionofMeromorphicFunctions(Mittag-Leffler Theorem)157 Problems162 Chapter 8.MultivaluedFunctioDs 6-1Introduction 6-2Examples ofInverse FunctionsWhichAreMultivalued 6-3The LogarithmicFunction 6.4Differentiability ofMultivaluedFunctions. 6-5Integration around aBranch Point 6-6Position of BranchCut 6-7TheFunction w=3+ (3 2 - 1 ) ~ i 6-8LocatingBranch Points. 6-9Expansion ofMultivaluedFunctions in Series. 6-10ApplicationtoRootLocus Problems 169 169 170 176 177 180 ISS 185 186 188 190 197 CONTENTSXV Chapter7.Some UsefulTheorems.201 7-1Introduction201 7-2Properties ofRealFunctions.201 7-3GaussMean-valueTheorem(andRelated Theorems).205 7-4Principle oftheMaximum andMinimum207 7-5AnApplicationtoNetwork Theory.208 7-6The IndexPrinciple211 7-7Applications of theIndexPrinciple,NyquistCriterion213 7-8Poisson'sIntegrals.215 7-9Poisson'sIntegralsTransformed to the Imaginary Axis220 7-10Relationships betweenRealandImaginaryParts,forRealFrequencies223 7-11Gainand AngleFunctions229 Problems231 Chapter 8.Theoremson RealIntegrals234 8-1Introduction234 8-2PiecewiseContinuousFunctions ofaRealVariable234 8-3Theorems and Definitions forRealIntegrals236 8-4Improper Integrals.237 8-5AlmostPiecewiseContinuous Functions240 8-6Iterated Integrals ofFunctions ofTwoVariables(Finite Limits)242 8-7Iterated Integrals ofFunctions ofTwoVariables(InfiniteLimits)247 8-8Limit under the Integral forImproper Integrals.250 8-9MTest forUniformConvergenceofanImproperIntegraloftheFirst FUnd.251 8-10A Theorem forTrigonometricIntegrals.252 8-11TwoTheorems onIntegration overLarge Semicircles.254 8-12Evaluation of Improper RealIntegrals by Contour Integration.259 Problems263 Chapter9.TheFourier Integral.268 9-1Introduction268. 9-2Derivation of the Fourier Integral Theorem.268 9-3SomeProperties of the Fourier Transform.273 9-4Remarks aboutUniqueness and Symmetry273 9-5Parseval's Theorem279 Problems282 Chapter 10.The LaplaceTransform285 10-1Introduction285 10-2TheTwo-sidedLaplaceTransform285 10-3Functions ofExponentialOrder.287 10-4TheLaplaceIntegral forFunctions ofExponential Order288 10-5ConvergenceoftheLaplaceIntegral fortheGeneralCase289 10-6Further Ideas aboutUniformConvergence.293 10-7Convergenceofthe Two-sidedLaplaceIntegral295 10-8TheOne- and Two-sidedLaplaceTransforms.297 10-9Significance of Analytic Continuation in Evaluating the Laplace Integral298 10-10Linear Combinations ofLaplaceTransforms.299 10-11LaplaceTransforms ofSomeTypical Functions300 10-12Elementary Properties of F(8)306 xviCONTENTS 10-13The Shifting Theorems309 10-14Laplace Transform of the Derivative of f(t)311 10-15Laplace Transform oftheIntegral of aFunction312 10-16Initial- and Final-value Theorems314 10-17Nonuniqueness of Function Pairs forthe Two-sided LaplaceTransform315 10-18The InversionFormula318 10-19Evaluation of the InversionFormula322 10-20Evaluating theResidues(TheHeavisideExpansion Theorem)324 10-21Evaluatingthe InversionIntegralWhen F(s)Is Multivnlued326 Problems.328 . Chapter 11.Convolution Theorems.336 11-1Introduction336 11-2Convolution in the tPlane(Fourier Transform)337 11-3Convolution in the tPlane(Two-sidedLaplace Transform)338 11-4Convolution in the t Plane (One-sided Transform) .342 11-5Convolution in the s Plane(One-sided Transform).343 11-6Application ofConvolution in the sPlane toAmplitudeModulation347 11-7Convolution in the 8Plane(Two-sidedTransform)349 Problems.350 Chapter 12.Further Properties ofthe LaplaceTransform353 12-1Introduction353 12-2Behavior of F(s)at Infinity.353 12-3Functions of Exponential Type357 12-4A SpecialClass ofPiecewiseContinuous Functions362 12-5Laplace Transform of the Derivative of a Piecewise Continuous Function of Exponential Order.367 12-6Approximation of f(t)by Polynomials.370 12-7Initial- and Final-value Theorems372 12-8ConditionsSufficient toMake F(8)aLaplace Transform.374 12-9Relationships betweenof f(t)and F(s).376 Problems.378 Chapter 13.Solution ofOrdinary Linear Equations with Constant Coefficients381 13-1Introduction381 13-2Existence of aLaplace Transform Solution foraSecond-order Equation381 13-3Solution ofSimultaneous Equations.384 13-4TheNatural Response388 13-5Stability.390 13-6The Forced Response.390 13-7illustrative Examples.391 13-8Solution forthe Integral Function395 13-9Sinusoidal Stee.dy-stateResponse397 13-10ImmittanceFunctions.398 13-11Which Is the Driving Function? .400 13-12Combination of ImmittanceFunctions400 13-13Helmholtz Theorem403 13-14Appre.ise.lof the Immittance Concept and theHelmholtzTheorem.405 13-15The System Function406 Problems.407 CONTENTSxvii Chapter 1'- Impulse Functions.410 14-1Introduction410 14-2Examples of an Impulse Response410 14-3ImpulseResponsefortheGeneralCase.412 14-4Impulsive Response415 14-5Impulse Excitation Occurring at t=Tl418 14-6Generalization of the "Laplace Transform" of the Derivative419 14-7Response to the Derivative and Integral of an Excitation422 14-8The Singularity Functions424 14-9Interchangeability of Order ofDifferentiation and Integration425 14-10Integrands with Impulsive Factors.426 14-11ConvolutionExtended to Impulse Functions428 14-12Superposition430 14-13Summary431 Problems.433 Chapter Iii.Periodic Functions435 15-1Introduction435 15-2LaplaceTransform ofaPeriodicFunction.436 15-3Application to the Response ofaPhysicalLumped-parameter System.438 15-4Proof That .c-1[P(s»)Is Periodic.440 15-5The CaseWhereH(s)Has aPole at Infinity441 15-6Illustrative Example.~ Problems.444 Chapter 16.TheZTransform445 16-1Introduction445 16-2The Laplace Transform of /*(t)446 16-3ZTransform of Powers of t.448 16-4ZTransform of aFunctionMultipliedhy ca. .449 16-5The Shifting Theorem.450 16-6Initial- and Final-value Theorems450 16-7The Inversion Formula451 16-8PeriodicProperties of F*(s),andRelationship to F(s)453 16-9TransmissionofaSystemwithSynchronizedSamplingofInputand Output.456 16-10Convolution.457 16-11The Two-sided ZTransform.458 16-12Systems with SampledInput andContinuous Output459 16-13Discontinuous Functions462 Problems.462 Appendix A465 Appendix B•468 Biblioll'aphy •469 Index.471 COMPLEX VARIABLES AND TIIE LAPLACE TRANSFORM FOR ENGINEERS CHAPTER1 CONCEPTUALSTRUCTUREOFSYSTEMANALYSIS 1-1.Introduction.Itisworthwhileforaseriousstudentofthe analytical approach to engineering to recognizethat oneimportant facet ofhiseducationconsistsinatransitionfrompreoccupationwithtech- niques of problem solving, with which he is usually initially concerned, to the more sophisticated levels of understanding which make it possible for him to approach asubject morecreatively than at the purely manipula- tive level.Lack of adequate motivation to carry out this transition can beaseriousdeterrenttolearning.Thischapterisdirectedatdealing withthismatter.Althoughitisassumedthatyouarefamiliarwith theLaplacetransformtechniquesofsolvingaproblem,at leasttothe extentcoveredinatypicalundergraduatecurriculum,itcannotbe assumedthatyouarefullyawareoftheimportanceoffunctionsofa complexvariableorofthewideapplicabilityofthe Laplacetransform theory. Sincemotivation is the primary purposeofthis chapter, forthe most part weshall make little effort to attain aprecision of logic.Our aim is to form abridge between your present knowledge,which is assumed to be attheleveldescribedabove,andthemoresophisticatedlevelofthe relativelycarefullyconstructedlogicaldevelopmentsofthesucceeding chapters.In this firstchapter webriefly useseveral concepts whichare reintroducedinsucceedim:;chapters.Forexample,wemakefreeuse olcomplexnumbersinChap.1,· althoughtheyarenotdefineduntil Chap.2.Presumablyastudentwithnobackgroundinelectric-circuit theoryorotherapplicationsofthealgebraofcomplexnumberscould studyfromthisbook;buthewouldprobablybewelladvisedtostart with Chap.2. Most ofChap.1 is devoted to areview of the roles played by complex numbers, the Fourier series and integral, and the Laplace transform in the analysis oflinear systems.However,the theory ultimately to bedevel- oped in this book has applicability beyond the purely linear system,par- ticularlythroughthe various convolutiontheoremsofChap.11and the stability considerations in Chaps.6,7,and 13. 1-2.ClassicalSteady-stateResponseofaLinearSystem.Abrief summaryoftheessenceofthesinusoidalsteady-stateanalysisofthe 1 2COMPLEXVARIABLESANDTHELAPLACETRANSFORM responseofalinearsystemrequiresapredictionoftherelationship between the magnitudes Aand B and initial angles aand {3for two func- tions such as Va=Acos(ClJt+ a) Vb=BCOS(ClJt+ {3) (1-1) where Va'for example, is adriving function* and Vbis aresponse function. From asteady-state analysiswelearn that it is convenient to define two complex quantities Va=Aei a (1-2) whichare related to each other through asystem functionH{jCIJ)by the equation Vb= H(jCIJ)Vo(1-3) H(jCIJ),acomplex function of the real variable CIJ,provides all the informa- tionrequiredtodeterminethemag- Lf1nituderelationshipandthephase r I+ }differencebetweeninputandoutput 110+",+q""sinusoidalfunctions.Presentlywe shallpointoutthatH(jCIJ)alsocom- pletelydeterminesthenon periodic FiG.1-1.A physical sysiem describedresponseofthesystemtoasudden by the function in Eq.(1-4). disturbance. In the example of Fig.1-1,the H(jCIJ)function is •jCIJRC H(}CIJ)=1_ClJ2LC+ jCIJRC(1-3a) H(jCIJ)=ClJRCei[r/2-tag-'wRC/(1--..·LCl)(1-3b) v' (1- ClJ2LC)2+ ClJ 2 R2C2 Equation(1-3a)emphasizesthefactthatH(jCIJ)isarationalfunction (ratio of polynOInials)of the variablejCIJ,and Eq. (1-3b)places in evidence thefactorsofH(jCIJ)whichareresponsibleforchangingthemagnitude and angle of V o , to give Vb.Evaluation of the steady-state properties of a system isusually in terms of magnitude and angle functions given in Eq. (1-3b),but the rational form is more convenient for analysis. This brief summary leaves out the details of the procedure forfinding H{jCIJ)fromthedifferentialequationsofasystem.It shouldberecog- nized that H (jCIJ)is a rational function only for systems which are described by ordinary linear differential equations with constant coefficients. 1-3.Characterization of the System Function as aFunction of aCom- plexVariable.Thematerialoftheprecedingsectionprovidesour first • Thetennsdrivingfunction,forcingfunction,andexcitationfunctionareused interchangeably inthis text. CONCEPTUALSTRUCTUREOFSYSTEMANALYSIS3 point of motivation for a study of functions of a complex variable.In the first place, purely for convenience of writing, it is simpler to write RCs H(s)= 1 + RCs + LCs2(1-4) whichreducestoEq.(l-3a)if wemake the substitution s= jlJJ.How- ever,whereverwewritean expressionlikethis,withsindicatedasthe variable,weunderstandthatsisacomplexvariable,notnecessarily jlJJ. Infact,throughoutthetextweshallusethenotations= tT+ jlJJ. Another advantage ofEq.(1-4)isrecognizedwhen it appears in the fac- tored form H(s)=(R/L)s (s- Sl)(S- 82) (1-5) Carryingtheseideasabit further,weobservethatthegeneralsteady- state-system response function can be characterized as a rational function H(s)=K-;.(_s _-_S-= .0.This permits definition of afunctionofu+ jw,whichbearsthesamerelationshiptova(t)e-VIas 'Oa(jw)bearstova(t),withtheadditionalstipulationthatva(t)isnow zero fort< o.Thus,wedefine Va(u+ jw)=fo 00vaCt)e-(o'+i"')1dt(1-25) and aformulacorresponding to Eq.(1-23a)can bederived,giving vaCt)e-V1=ir f-....Va(u+ jw)ei"'ldw •The Fourier integral forthe unit step, the functionwhichiszero fort< 0and1 fort> 0,is fo"if""dt=/0"(cos wt+ sin we)dt Neither cosinenor sinecan be integrated fromzero to infinity. CONCEPTUALSTRUCTUREOFSYSTEMANALYSIS whichis moreconvenientlywritten vaCt)=ir j: ..Va(q+ jw)e( f1 +i fO )Idw Similar expressions apply forVbCt),forwhichwehave Vb(q+ jw)=fo"Vb(t)e-(t1+i.. )'dt 9 (1-26) (1-27) astheFourier integral of Vb(t)e-"',where Vb(t)=0whent< 0;and also, in similaritywithEq.(1-26), Vb(t)= ~f"Vb(q+ jw)e( f1 +i fO )'dw(1-28) 2r_00 It isbeyondthescopeofthischaptertoshowthatVa(q+ jw)and Vb(q+ jw)bear arelationshipsimilar to Eq.(1-24),namely, Vb(U+ jw)=H(q+ jw)Va(q+ jw)(1-29) It now becomesapparent that Eqs.(1-25)through(1-29)arematerially simplified by regarding qand w as the components of the complex variable s,in whichcaseEqs.(1-25)and(1-26)become Va(s)=10"va(t)e- ol dt (1-30) va(t)=~hrVa(s)e"ds (1-31) wherethelastintegralisacontourintegralofthecomplexfunction Va(s)ealtakenoveraverticallineforwhichtherealcomponentqis constant.Thatis,onthecontourofintegration,s=q+ jwand ds= jdw.(ContourintegrationisthetopicofChap.4.)Asimilar pair ofequations applies to Vb(t),giving Vb(S)=10" vb(t)e-"dt Vb(t)=o ~ .[Vb(S)e"ds ""'"3]Br and Eq.(1-29)becomes (1-32) (1-33) (1-34) ThefunctionsVa(s)andVb(S)arecalledLaplacetransforms.The possibility of having prescribed initial conditions at t=0 is not admitted by Eq.(1-34);but it isarelativelysimplematter to show how this can behandled by adding another term,giving Vb(s)=H(s) Vo(s)+ G(8)(1-35) 10COMPLEXVARIABLESANDTHELAPLACETRANSFORM as the general expression V b (8).G(8) is a function of initial-energy terms. For details,you arereferred to Chap.13. In viewoftheimplicationsofEqs.(1-30)and(1-31),H(8}takeson addedsignificancewhentheFourier integralisextended to the Laplace integralformulation.UntilthisintroductionoftheLaplaceintegral, wehavebeeninterestedprimarilyinH(jw},althoughtheobservation has been made that considerable simplification ensues if H(jw) is regarded as a special case of H(s}, thereby making it possible for general properties ofsto be used in interpretation and designproblems in whichH(jw}is theprimaryfunction.Now,withthe Laplaceintegralformulationwe findH(8}appearing explicitly as a functionof8rather than of jw. In the developments of the last two sections wehave made freeuse of improper integrals.This fact points to another of the topics which must beconsidered,the question of properties ofthe integrand functionsthat will make the integrals exist.Perhaps more important is the fact that a formulalikeEq.(1-33)is useful only if it can be evaluated and if it can beinterpretedtodeterInineitspropertiesasafunction.Therefore, techniquesofevaluatingandmanipulatingimproperintegralsprovide one of the later objectives ofthis study. 1-'1.Frequency,and the Generalized Frequency Variable.TheFou- rierintegralcarriestheliInits- 00and00,whereintegrationiswith respect to the variable w,implying that we are interested in functions of w fornegativeaswellaspositivevaluesofw.Intheanalysisoftime responseofsystemsitiscustomarytocallw theangularfrequency, recognizing it as related to the actual frequency fby the simple formula w=2rf What,then,isthe physical meaning ofwand fwhen they become' nega- tivenumbers?Nophysicalinterpretationseemspossible,sincefre- quency is by definition a count of number of cycles or radians per second. The error is in calling f and w frequency;the' proper terminology is and Frequency=III Angular frequency=Iwl The alternative isto referto fasthe frequencyvariable,rather than frequency.However,the quantityw issomuch moreprevalentthan f in analytical work that weshallconsiderw to be the frequencyvariable. Sometimes in analysis it isconvenientto regard the Laplacegenerali- zationoftheFourierintegralasequivalenttoageneralizationofa sinusoidaldriving function.For example,in Eq.(1-23a)wemay think ofvo(t}asduetoasuperposition(viatheintegral)ofsinusoidalcom- ponents like CONCEPTUALSTRUCTUREOFSYSTEMANALYSIS11 which is essentially the cosine function.To clarify this statement, it can beshownthatforapracticalsystem,havingarealresponsetoareal excitation,1),,( -jw)istheconjugateof1),,(jw),andsoif A(w)isthe magnitude functionand a(w)isthe angle function, 1),,(jw)=A(w)e ia (.. ) and 1),,( -jw)=A(w)e- ia (.. ) in terms ofwhich the abovebecomes 2A(w)cos[wt+ a(w)]dw In the Laplace case,thecorresponding formulais v,,(t)=.2....(V,,(s)e"ds= fooV,,(u+ jw)e(ff+;..)1dw 27rJlBr27r- 00 which implies asummation of components like [V,,(u+ jw)e(ff+MI+ V,,(u- jw)e(ff-iW)I]dw=[V,,(s)e"+ V,,(s)eiljdw Thisfocusesattentiononeo ' insteadofe iw1 asthebasicbuildingblock. FIG.1-4.Plots ofthe function + Il'''),where 80""/TO+ j",.,forthree values of /T.and with "'. constant. The above can be made to look more like the previous caseby writing it in the form e"t[V,,(s)e iwt + V,,(s)e-i"'tj showingittobeasinusoidalfunctionmultipliedbyanexponential. * Examples ofthe specialcase + e-i... I ) are plotted inFig.1-4. • ThisistruebecauseinpracticalproblemaV.(')istheconjugateofV.(s). 12COMPLEXVARIABLESANDTHELAPLACETRANSFORM In view of the factthat eo'is ageneralization ofe iw "it iscustomary to callsthegeneralizedfrequencyvariable.Thisisofcourseincomplete consonancewith the previously observed factthat H(jw) can be general- ized by replacing it by H(s).In that casealso,the variable 8should be thought of as the generalized frequency variable, an idea which is implied byEq.(1-34).Thereisoneunfortunateconsequenceofthistermi- nology;thefrequencyvariablewistheimaginarycomponentofthe generalized frequency variable.It would be conceptually ~ o r esatisfying if w were the real part of 8.However, as a consequence of certain factors whichleadtosimplificationselsewhereinthetheory,thesubjecthas developed with t h i ~apparently anomalous situation. 1-8.Stability.Stabilityisoneofthe importantconsiderations inall problemsofsystemdesign.Thiscommentapplieswhetherthesystem islinearornonlinear.In fact,onewaytodeterminewhetherornota nonlinear system is stable is to consider that initial disturbances are small and to consider the system momentarily linear.In that case,there is no difference in the consideration of stability between a system that remains linearandonethatisbasicallynonlinear.Infact,everyunstable, physicallyrealizablesystemmusteventuallybecomenonlinearasthe responsecontinues to build up. Adetailedanalysisofsystemresponse,suchasisgiveninChap.13, showsthatthevaluesof8atwhichH(s)becomesinfinitecar!L the essential information as to whether or not the system is staple.Referring to Eq. (1-6), t h ~system is stable if the real parts of the numbers 81,82, etc., are nonpositiye.Thus,the questionofstability provides further reason tostudythevariouspropertiesofH(s).Twoimportantengineering techniquesfordealingwiththequestionofstabilityaretaken· upin Chap.6(the root locus)and inChap.7(theNyquist criterion). These methods of studying stability can be used,purely as techniques, withonly superficialknowledge;but theirjustifications aregrounded in quitesubtlepropertiesoffunctionsofacomplexvariable.Therefore, if asatisfying degree of understanding ofthe question of stability of both linear and nonlinear systems is to be acquired,there is no recourse but to becomeacquaintedwiththetheoryoffunctionsofacomplexvariable. 1-9.Convolution-type Integrals.Integrals ofthe form and arisefromsituationswhichareessentiallydivorcedfromthecomplex- functionviewpointofnetworkresponseorLaplacetransformtheory. For instance,the firstof these isthe result weget by applying the super- position principle to obtain the response ofalinear system.The second CONCEPTUALSTRUCTUREOFSYSTEMANALYSIS13 type of integral occurs in the theory of correlation.Similar integrals also occur in the theory of the responseof nonlinear systems. IntegralsofthistypebeararelationshiptoLaplaceandFourier transformsbyvirtueoftheirLaplaceandFouriertransformsbeing productsofthetransformsofthefunctionsappearingintheintegrand. Thispropertyprovides avehiclewherebythe Laplacetransformcan be brought intoplay in situations other than the basicone described in the bulk ofthis chapter. Oneexample,giveninChap.11,makesuseofaconvolution integral intransformfunctions,showinghowLa.placetransformtheoryis aplicableintheessentiallnonlinearulationand demoation. 1-10.Idealized Systems.In many practical situations, linear systems ofthetypesconsideredhereareparts oflargersystems.In thedesign oftheselargersystemsit isoftenconvenientto idealizethecomponent subsystems.Whenthisisdone,thecomponentpartsaredescribedby idealizedandangle(phase)responsefunctionsofrealfre- quency.In this discussion,it is not possible to generalize the aspects of all design problems in one sentence.However, it is generally true that an idealized response ischosen to givean adequate(and possibly optimum) timeresponseto adesiredsignal,whilerejectingunwantedsignals,and toprovideasystem that isstable.The followingtwo examples can be given:Incommunicationsystemsfilteringisusedtoprovideanintel- ligible signal in the presenceofnoise; and in control systems an accurate reproductionofacontrolsignalisrequired.Filtersandcorrective systems (electrical networks, and sometimes mechanical or other systems) are encountered in allcases.Becausethe time response,or an estimate thereof, is the usual end result, if we attempt to think in terms of idealized frequency-responsefunctions,alinkbetweentime- andfrequency- responsefunctionsisessential.ThislinkisprovidedbytheFourier integral theorem. OnethennaturallyaskswhytheFourierintegraltheorem,whichis basically atheoremrelatingrealfunctionsofrealvariables,isnot suffi- cient.As a partial answer to this question we submit the following ideas: In the first place, once the idealized characteristics of a system component (filter,forexample)havebeendecidedupon,thedesignerisfacedwith the problem ofcreating aphysically realizable device which will approxi- matetheideal.Thisisthesynthesisproblem.Wehaveseenthat electricnetworks,forexample,arecharacterized by functionsofacom- plex variable,and hence atranslation of idealized response functions into realizablefunctionsinevitablyinvolvesfunctionsofacomplexvariable. Also, once a realizable system has been designed, an ana.lysis of its specific time-responsecharacteristicsrequiresthesolutionofintegrodiiJerential 14COMPLEXVARIABLESANDTHELAPLACETRANSFORM equations.Then the Laplace integral and transformbecome important andaremorecloselyrelatedtotheFourierintegraltheoremthanthe various other methods available forsolving thesesame equations. 1-11.Linear Systems with Time-varying Parameters.The emphasis in this chapter on linear equations with constant coefficients should not be construedtoimplyneglectofsysteIDSwithtime-varyingparameters. Such systeIDSare important, and many ofthem fallwithin the realmof linearsysteIDS.Theyareomittedfromdetailedconsiderationhere because the treatment ofthis chapter is basically superficial,and to add this further complication would magnify the appearance of superficiality whilecontributing little to the main objective. 1-12.OtherSystems.Systemsinwhichtimeistheindependent variablearecertainlyimportantandprovidethemainvehicleforthe examplesinthistext.However,theydonotexhaustthepractical applications of the material presented.Many field problems yield linear equations,and it isshownin Chap.3that the theoryoffunctionsofa complexvariableisdirectlyapplicabletocertainfieldproblemsintwo dimensions.Also,the linear antennaisanotherimportantapplication. When applied to antennas,the Fourier integral plays arolevery similar to that played in the theory of the time response of linear systems.Thus, thematerialpresentedinthistextisapplicableinseveralareasnot illustrated in this introductory chapter. PROBLEMS 1-1.Obtain the functionH(s)forFig.PI-I, assuming that displacementxis the driving functionand II is the response.. Sliding block " FIG.P1-1 Dashpot . 1-1LReferringtoFig.P1-2,letIIIbethedrivingfunctionandIIItheresponse. Obtain an expression forH(,) forthis system. CONCEPTUALSTRUCTUREOFSYSTEMANALYSIS15 C-l FIG.P1-2 1-8.An inductor of Lhas asaw-tooth currentflowing,of the formshown in Fig. P1-3. (a)Write the Fourier series forthis current. (b)From this,usingH(je.»on each term,obtain theFourier series forthe voltage across the inductor. (c)Fromthedifferentialequation"- Ldi/dt,determinethewaveshapeofthe voltage,andfinditsFourierseries,usingtheformulafortheFouriercoefficients. Compare the result withpart b. '-...~ / ~ o: ~T ~ T (sec) FIG.P1-3 1-4.A periodic functioncan be represented by aFourier series in either of the fol. lowingequivalent forms: lA ..cos(nw.t+ a .. ) n-O whereA ..anda ..arerealandC..iscomplex.ObtainaformulaforC..in termsof Aftand a". R FIG.P1-5 1-5.Consider the circuit ofFig.P1-5, forwhich the excitation is the voltage pulse t 1 It I < 1 are given,and letfl(j",)andS(j",)bethe corresponding functionsobtained fromthe Fourier integral. 18COMPLEXVARIABLESANDTHELAPLACETRANSFORM (a)Find 'J(j... )and g(j... ). (b)Find/_..../(.,.)g(t- .,.)d.,.and/_....g("')/(t- .,.)d.,.,showing they are the same function. (c)EvaluatetheFourierintegralforthe functionobtainedinpartb,andcheck whether or not this result is identical with 'J(j... )g(j... ). 1-1&.The functions I(t)and g(t)are zerofor negativet,and fort~0 are given by I(t)=8- 1 g(t)=e ' Let F(s)and G(s)be the respective functionsobtained fromthe Laplace integral. (a)Find F(s)and G(s). (b)Find /o'/(.,.)g(t- .,.)d.,.a'ld /0' g(T)/(t- .,.)d.,.,showing that they are the same. (c)Find the functionobtained fromthe Laplaceintegral of the functionobtained in part b,and compare the result with F(s)G(s). CHAPTER2 INTRODUCTIONTOFUNCTIONTHEORY 2-1.Introduction.Thetheoryoflinearsystems,particularlywhen castintermsoftheLaplacetransform,reliesheavilyonthetheoryof functionsofacomplexvariable.Abriefinsightintothisdependence wasgiveninChap.1.Inthenextfewchaptersweshalldevelopthe theory offunctionsofacomplex variableto provide thebackground for furtherstudyoflinearsystemsandrelatedsubjects,particularlythe Laplace transform and convolution integrals. Beforecontinuing,aword about howweshall approach the subject is inorder.Weshallnotproceedaswouldamathematician,whowould placeemphasisonrigorandgeneralityofthetheorems.However,it willbethegeneralitymorethantherigorthatweshallgiveup.In mathematicsoneoftheobjectivesisalwaystoprovetheoremsforthe most general cases possible.For us to do this would be awaste of time, looking as wearetoward the utilitarian value of the subject, because the most generalconditions are not needed.By this wemean that youwill encounter most of the standard theorems, but applied to relatively simple cases.There willbenosignificantlossofrigor,and thereforethework shouldbesatisfying to the thoughtful reader.However,becauseofthe reduction in generality, you should not regard this work as a mathematics course in functionsofacomplex variable. At the beginning weshall assumethat you are familiarwith algebraic manipulation of complex numbers, but the subject will be reviewed.You sho,uldunderstandthatacomplexnumberAisanorderedpairofreal numbersAl and A2whichcan bewritten symbolically A=(A I ,A 2 )(2-1) A second complexnumber may be designated by B=(B I ,B 2 ) Using these as examples, the algebraic operations are defined as follows: 1.Identity A=B if and only if Al = BI and A, =B 2 • 19 (2-2) 20COMPLEXVARIABLESANDTHELAPLACETRANSFORM 2.Addition (2-3) 3.Multiplication AB =(AlBl- A 2 B"AlB, + A2Bl) (2-4) It isleftasanexerciseforyoutoshowfromthesedefinitionsthat additionandmultiplicationobeythecommutative,associative,and distributive lawsofalgebra. 4.Division.Inasystemconsistentwithrealnumberswecannot definedivision independently.Weshall want C,where A C="B tobethenumber suchthat whenmultipliedby Bit willgiveA.It is then possibleto prove ~= (AlBl + A2B!A2Bl- AIB2)(2-5) BB ~ + B ~,B ~ + B ~ If acomplexnumber has the special form (R,O) it is said to be real and wecan write R= (R,O) Thus,wemakeadistinctionbetweenarealnumberRandacomplex number which hasthe real valueR. Another frequently occurring formis (0,1) This is said to be an imaginary number, but as yet we have introduced no symbol forit. As aresult ofthe above terminology,it hasbecomethe custom,given A=(Al,A!),to callAl the realpart(ortherealcomponent)and tocall A2theimaginarycomponent.Also,thenumber(O,A!)iscalledthe imaginary part ofA. It isconvenienttohaveanotationtodenoterealandimaginary components.For this weuse Al=Re A A2=1m A Here are three complex numbers ofgreat importance: ° =(0,0) 1=(1,0) j=(0,1) INTRODUCTIONTOFUNCTIONTHEORY21 The complex numbers 0 and 1 play the same roles in the operations with complexnumbersasdotheircounterpartsinrealnumbers.Anumber added to 0 is unchanged,and anumber multiplied by 1 isunchanged. The specialimaginary number j(written iin mathematics literature) has no counterpart in real numbers.From the rule of multiplication note that jA=(0,1)(AI,A2)=(-A 2 ,A 1 ) Thus,multiplyingacomplexnumberbyjinterchangesitsrealand imaginary componentswithasubsequentsignchangeofthenewreal component.In particular note that jj = (-1,0)(2-6) Thenumberjisimportantbe- causeitprovidesahandywayto write a complex number.Byapply- ing the rules ofalgebra weget A=(A 1 ,A 2 ) =(AI'O)+ (0,A2) ~ OJ c "lib-------- .§A2 =(At,O)+ (0,1)(A2'0)FIC{.2-1.Geometricinterpretation of a =Al + jA2(2-7)complexnumber. Because a complex number is an ordered pair of real numbers, it can be represented pictorially as apoint in aplane,as shown in Fig.2-1.This portrayal suggests defining the magnitude and angle of acomplex number as follows: magnitude angle (2-Sa) (2-Sb) * It willsometimesbeconvenienttodesignatetheserespectivelybythe notation IAI= mag A a=angA These two quantities (magnitude and angle)can be interpreted geometri- cally as the polar coordinates ofapoint in aplane. In writingacomplexnumberit issometimesconvenienttodrawon this geometrical interpretation and to write A=lAlla (2-9) However,IAIand ado not have quite the fundamental significance of Al and At.It wouldbeinconvenientto useIAIand ato defineacomplex • In much of the literatureIAI iscalledthe modulm and athe argument, in which case the abbreviations are mod Aand srg A. 22COMPLEXVARIABLESANDTHELAPI..ACETRANSFORM numberbecauseofthemultivaluednessofajlAlla= lAlla + 211",for example.Thus, foragivencomplexnumber,the angleis not aunique number. It is apparent fromthe geometrical interpretation that At=IAIcos a AI=IAIsin a Also,it takes only alittle trigonometry to show that IABI=IAIIBI ang AB=ang A + ang B and I ~ l = ~ A ang B =angA- angB (2-10) (2-11) (2-12) Inviewoftheserulesformultiplicationanddivision,aconsistent definition can be given for the root of acomplex number.We shall write (A)J.iasthe symbolforthenumberwhichmultipliedby itselfgivesA. Thus, if AJ.i=IBI/13,and A =lAlla, it followsthat and therefore IB12/213=lAlla B=ViAl However, we note that lAlla=lAlla +211",and therefore a second value of angle- is possible.Since a' 13=-+11" 2 this possibility of adding 11"corresponds to the usual sign ambiguity of the square root.Athird value correspondingtolAlla=lAlla + 4Jrisnot obtained,becausea/2+211"isgeometricallythe same as a/2.Thus,in similarity with real numbers,AJ.ihas two roots.Likewise,A14has three roots: which correspond to the three geometrically equivalent values lAlla=lAlla +211"=lAlla +4Jr INTRODUCTIONTOFUNCTIONTHEORY Finally, forthe generalcase,A 1/..has ndistinct roots: 23 k=0,1,2,...,n - 1(2-13) Weconcludethisintroductionbymentioningthecomplexconjugate of A, which is written A and definedby A = (Al,-A.) =Al- jA,(2-14) The complex conjugate (or conjugate)of anumber is obtained by chang- ing the signofthe imaginary part. From the rules ofalgebra it followsthat and also A+B=A+B AB=AB m=! IAI2=AA ReA=ReA=~ ( A+ A) - 1- ImA=-1m A=2] (A- A} (2-15a) (2-15b) (2-15c) (2-16a) (2-16b) (2-16c) Certain inequality relationships are important in the subsequent work. From the geometryofFig.2-1it isevident that IReAI~IAI 11mAI~IAI (2-17a) (2-17b) NowconsiderIA+ BIZ,which,inaccordancewithEqs.(2-15a)and (2-16a),canbewritten IA+ BIZ= (A+ B)(A + B) =AA + BB + AB + AB =IAII+ IBI2+ 2Re(AB) i;In the last lineabove,Re (AB)maybenegative.The right-handside l willbeincreasedorunchangedif wewriteIRe(AB)Iordecreasedor f unchanged if wewrite-IRe (AB)I.Therefore,it followsthat IAI'+ IBI2- 21Re(AB)!~IA+ BI2~IAI2+ IBlz+ 21Re(AB)I ;Al80,fromEq.(2-17a) IRe(AB)I~IABI=IAIIBI=IAIIBI ,and thereforethe previous inequality simplifies to IAII+ IBlz- 21AIIBI~IA+ BIZ~IAI2+ IBlz+ 21AIIBI 24COMPLEXVARIABLESAND~ H ELAPLACETRANSFORM Finally,by taking square roots weget IIAI-IBII ~IA+ BI~IAI+IBI (2-18) This result isan analyticalstatement ofthe factthat the lengthofone side of a triangle is less than the sum of the lengths of the other two sides but greater than their difierence. 2-2.Definition of a Function.One of the most important concepts to be established is the idea of a complex number being a function of another complex number.Let the symbol 8= CT+ jw (2-19) represent a complex number, where CTand w each may have any real value between negative and positive infinity.(Acomplexnumber designated w Analytic·geometry plane •(u,W) CT (a) w-lmB Complex plane •(B-U+jW) CT-ReB (6) FIo.2-2.Comparison of analytic-geometry plane andcomplexplane. in this way is commonly called a complex variable, although in reality it is nomore"variable"thananyothercomplexnumberdesignatedbya letter symbol. )You are familiar with the use of an "analytic-geometry" planeforplottingtherelationshipbetweentwonumbers(variables) such as CTand w.Such a plane is shown in Fig. 2-2a, in which a representa- tivepoint has coordinates(CT,W)and the axesare labeled accordingly. Thecomplexnumber8,asdefinedbyEq.(2-19),providesaslightly different way to represent apoint in the plane.Figure 2-2bshows what weshallcallthecomplex8plane.Geometricallyit isthesameasthe analytic-geometryplane,butphilosophicallyitisquitedifferent.In the 8planethe axesare labeled"real" and"imaginary,"and atypical pointislabeled with asinglesymbol,namely,8.Asyouread this,you should begin to acquire a feeling for the idea of using one symbol t.orepre- sent tworealvariables. In the subsequent developments we shall have much use for the idea of a complexplane,but occasionallyweshallrelateitbackto the analytic- geometryplaneforinterpretations.Meanwhile,eventhoughtheaxes may be labeled CTand w,these symbols willmeanRe 8and 1m 8. INTRODUCTIONTOFUNCTIONTHEORY25 Now weareready to introduce the notion of afunctionofacomplex variable.In additiontothe8plane,imagineasecondcomplexplane, which we shall call the wplane.Let whave the form w=u+jv(2-20) and suppose that a rule is stated whereby for each point in the 8plane (or portion thereof)auniquepoint is specifiedinthe wplane.We can say that w is afunctionof8,and wemay indicate that fact symbolically by writing W =f(8) In this definitionof afunctionweunderstand that foreach point in the 8planethereisonlyonepointtocorrespondtoit inthewplane.In other words, when we say function we shall understand the word to mean a single-valuedfunction.Atalatertimeweshallbe interestedin multi- valued"functions," but forthe present they will be avoided. Youshouldunderstandthoroughlythat the wplaneisgeometrically similar to the 8plane,differingonly inthe symbolusedtodesignateit. By this wemean that the w plane is also ageometric idea forportraying acomplexvariableandthatitalsoisquitesimilartotheanalytic- geometry plane ofthe pair of realvariables uand v. To pursue further the idea of a function of acomplex variable, consider the particular case W=8'(2-21) Notethatthereisnoquestionaboutthemeaningofthis,since8 2 has the meaning (8)(8),and multiplication has been defined.Thus, foreach point in the 8plane or w=(0- + jw)(o- + jw) =0- 2 - w 2 + j20-w U=0- 2 - w 2 V= 20-w (2-22) (2-23) The first idea you should get from this example is that a formula such as (2-21)does give a rule for determining points in the w plane to correspond to points in the 8plane.Equations(2-23)actually givethe rectangular coordinates of the w-plane points in terms of the rectangular coordinates ofthes-planepoints.Equation(2-21)issimplertowritethanEqs. (2-23),but they are geometrically equivalent. Note that Eq.(2-21)is only one of many functions that can be defined throughnothingmorethantherulesfor'addition,multiplication,and division.Additional functions like w=I+8W=8 1 1 W=- 8 W=8+8'etc. 26COMPLEXVARIABLESANDTHELAPLACETRANSFORM can be constructed.All that we require at this point is that the formula tellushow,givenavalueofs,thecorrespondingvalueofwshouldbe determined.SeveralofthesefunctionsareillustratedinFig.2-3.A generalizationoffunctionsconsistingoflinearcombinationsofpowers of 8can be written (2-24) wherethe a'sarecomplexconstants and ncanbeanypositiveinteger. The notation P(8) implies that the function in this case is apolynomial in 8.A further generalization ofthe functions we are prepared to deal with now is obtained if we have asecond polynomial and then let wbe the ratio P(s)ao+als+a2B 2 +•••+a n 8" w=-Q(-s)='b o '---+7----;b--=IS-+--i--c b ,.:2'-;;S2:-+-:-.-.-.-+--;---'-b"::'ms-m (2-25) You should have no difficulty in understanding that when the a's and b's are all known each valueofs gives avalue ofw which can be calculated. wplanes . w III) ~ ~ w \1........- iii FIG.2-3.Some examples of functionsof 8. We shall seldom actually make such calculationa numerically; in fact one ofour objectives is to get interpretations and meanings out of such func- tions without making calculations,or with aminimum ofcalculations. 2-3.Limit,Continuity.So farwehavereviewedthe algebraofcom- plexnumbers,andintroducedtheconceptsofacomplexvariableand functionsthereof.Wefoundthatrelativelysimplefunctionscouldbe definedwholly on the basis of algebraic operations of addition and multi- plication.Eventuallyweshalldefinemany other functions,but at this pointwehavedoneaboutallwecanwithoutgoingintotheideasof calculus. INTRODUCTIONTOFUNCTIONTHEORY Tocontinue,wenextexaminetheconceptsoflimitandcontinuity. Consider afunction W=f(8) and allow 8to approach a number 80along a line such as a in Fig. 2-4.In the functionplane w willingeneralapproach apoint labeled w ~ ,along a linea'.Nowsupposethatwalwaysapproachesw ~ ,regardlessofthe direction in which8approaches 80.If this isthe case,wesay w ~=limf(8)(2-26) Intheaboveweusethesymbolw ~ ,ratherthanwo,becausethe limit canexistevenwhenf(80)doesnotexist;andweshallreserveWoasa symbol for f(so). s plane wplane FIG.2-4.Geometric interpretation ofafunctionapproaching alimit. B planewplane (a) (b) FIG.2-5.Definition of limit in terms of Eand 8 neighborhoods. In precise mathematical language,wedefinethe limit w ~as existing if, when given a small arbitrary positive number E,it is possible to find a num- ber6 such that when If(s)- w ~ 1< E 0< Is- sol+ jp sin1/»=1 Whenthemultiplication iscarried out and realand imaginary parts are equated,weget pUcosl/>-pvsinl/> =l-u pvcosI/>+ pu sinI/>=-v (3-29) " ",,\' .... /cr--1.5 I I I CONFORMALMAPPING67 cr w-plane FIG.3-18.Transformationw=1/(8+ 1)forrectangularcoordinates in the8 plane. Squaring eachoftheseequations and adding the results gives (u 2 + v2)(1- p2)- 2u+ 1=0 and completing the square in ugives (3-30) This equation describes the family of loci in the wplane for circles of con- stant pinthe8plane,forallbut thedegeneratecasep=1.For that 68COMPLEXVARIABLESANDTHELAPLACETRANSFORM case the original equation shows that the locus is the vertical straight line Examples oftheselociare shownby the solid linesofFig.3-19. • plane wplane , , \ \ \ \ , 2 -1 -2 ,.",-- --....., .... "•--150" 'y! \ \ I I I I , I I 1"- ,/ .-150· -- .. ' '" 2 FIG.3-19.Map of the function w- 1/(8 + 1),using polar coordinates in the 8 plane. Lociinthe wplane corresponding to radiallinesofconstantc/Jin the 8 plane are obtained by dividing the first of Eqs.(3-29)by the second, giving ucotc/J- v=u_-_l v cotc/J+ uv and this inturn becomes u 2 - u+ v'- v cotc/J=0 Completing the square in both u and v then gives (u- ~ r+ (v- c o ~~ r=(2 S i ~c/Jr (3-31) Thusit isfoundthattheradialstraightlines(c/J=constant)inthe8 plane go into circles in the w plane.When sin c/J=0,we get a degenerate case;but this isthe realaxisin the 8plane,and it hasbeen shownthat CONFORMALMAPPING 69 thisgoesintotherealaxisofthewplane.Thus,whensin,p= 0,the locusis 11=0 Loci for constant tPare shown by the dashed circles in Fig. 3-19.Here wemaketheinterestingobservationthat eachofthesecirclesisinter- preted in two parts, an upper part and alower part, forwhichthe angle designations differ by 180°.This is a natural consequence of the fact that each circle isthe trace ofaradial line passing throughthe originofthe 8 plane; and of course the angular designation for such aline changes as the origin is passed. In the w plane weagain findthe circle into which is Inapped the right halfofthe8plane.Thisisthecirclecarryingthedesignation± 90°, whichofcourse isthe same as labeling it (J'=0,as in Fig.3-18. This treatment ofEq.(3-27)leads to an interesting generalconclusion about the function 1 W=- 8 Let 8describeacircledefinedby 8=A+Re;' where the complex Aand real R are constants and 8 is variable.Then we can write 1 w= A+ Re;' 11 =it 1 + (RIIA J)e;('-a) wherea= ang A.This isofthe form where 11 W=A8'+1 8'- ~e;('-a) -IAI (3-32) WerecognizeEq.(3-32)as being similartoEq.(3-27),and it isrecalled fromthe analysis ofthe latter equation that circles centered at the origin in the 8plane go into circles in the w plane.Therefore,Eq.(3-32)yields acirclein the wplane when8'describesacirclecenteredat the origin. Furthermore,wehave definedatin such away that it iscentered at the origin when 8describes the prescribed circle.It is concluded that w=1/B traces out acirclewhenever 8followsacircular path. 70COMPLEXVARIABLESANDTHELAPLACETRANSFORM 3-10.The Bilinear Transformation.In Sec.3-3it ismentionedthat the function 8- 1 10=-- 8+1 (3-33) has important applications in the analysis of linear systems.Because of this importance, we now give it abrief consideration.Equation (3-33)is aspecial caseofthe generalbilinear function as + b W=cs+d It ispreferable to study Eq.(3-33),and fromit wecan learn all weneed to knowabout the generalcase.Thetreatment of the previous section servesasthepointofdeparturebecause,aswaspointedout there,Eq. (3-33)can be written 10=1- _2_ s+1 We shall considerthe map ofrectangular coordinates inthe right half ofthe8plane.Figure3-18showsthetransformationoftherighthalf planebythe function1/(s + 1).Wemerelytakethemirrorimageof this transformation, scaled by afactor 2,and translate it one unit to the right.The resultisshown in Fig.3-20a.From Eqs.(3-28)weget the equationsofthelociby changingsign on u and v and multiplying them bythefactor~andbysubtracting1fromutoperformtherequired translation to the right.The results are (v- ~ r+ (u- 1)2= ar v2+ (u__u )2=(_1 )2 u+lu+l (3-34) Aparticularly importantportrayalofthistransformationisobtained by usingpolarcoordinatesinthe10plane.In ordertoobtainthecor- responding fociinthe 8plane,writeEq.(3-33)in the inverse form 1+102 s=--=-1 1 - 101 + (-10) The term 2 1 + (-10) is like Eq.(3-27),but with-10 in place of 8.Thus, to get the loci of the presentsplane,wetake the lociofthe10planeof Fig.3-19,withascale factorof2,andwithallanglelabelschangedby±r (toaccountfors w 2 1 -- - r - - - - - ~ f-.- -- ----- o12 f-.- -- ----- -u- CONFORMALMAPPING -21-.- -- ----- splane 3 2 1 (a)Rectangular coordinates in the s plane O H l - ~ ~ ~ - + - + - - - - ~ - -1 -2 -3 (b)Polar coordinates in thew plane v u v u wplane FIG.3-20.Transformation duetothe bilinear functionw=(8- 1)/(8+ 1). 71 72COMPLEXVARIABLESANDTHELAPLACETRANSFORM beingreplacedby-w).Thenthepatternisshiftedtothe leftaunit distance,withtheresult .showninFig.3-20b.Thismapisgivenfor Iwl~1. The above manipUlations of change of scale and shifting can be applied toEqs.(3-30)and(3-31)togetthes-planelociinFig.3-20b.Inso doing wereplace uby(0"+ 1)/2, v by ",/2,pby r,and q,by 8,where we arenowusingthepolar coordinates w= rei' in the wplane.The results,obtained fromEqs.(3-30)and(3-31),are, respectively, ( 1 + r2)2(2r)2 0"--- +",2=-- I- r21- r2 0"2+ ('"_ cot8)2=(_._1_)2 sm8 (3-35) Apparently this transformation also takes circles into circles,as indeed wecanseebyrecallingthatithasbasicallythesametransformation properties asl/s. We conclude this section with abrief discussionofthe general case as+ b w=cs+d whichcan be written _1(+be-ad) w- - a ecs+ d Certainly, if s describesacircle,sodoeses+ d.Frompreviousdiscus- sions,wealsoknowthat be- ad es+ d describesacircle,andconsequentlythew-planelocuswillbeacircle. Thus,the generalbilinear transformation carries circles into circles. WecanalsocastthegeneralcaseinaformrelatedtoEq.(3-33)by writing w=2 ~[(ad+ be)+ (ad- be)~ ~ j ~ ~ :~~ ] Thesecondtermisaconstantmultipliedby(s'- l)/(s' + 1),where s'=(e/d)s,andthistransformationhasbeentreated.The otherterm in the above equation is merely aconstant.Thus wesee how with suit- able change of variable the general case can be obtained fromthe specific one. CONFORMALMAPPING73 3-11.Conformal Mapping.In all the foregoing examples it is observed that inmost casesmutuallyperpendicular lines in the 8plane transform into mutually perpendicular lines in the wplane.However, we find there are some points in these illustrations where this is not true, the point 8=0 forthe functionw=8 2 beingoneexample.Here themutually perpen- dicular real and imaginary axes in the s plane transform into lines in the w plane intersecting at 180 0 ,as showninFigs.3-3and 3-4. Another property exhibited in most instances is that asmall geometric figure,like acurvilinear rectangle formedby four coordinate lines,trans- formsintoasimilarfigureinthewplane.Thecorrespondingareas labeledAand A' inFigs.3-3and 3-5are illustrations.However,again usingw= S2astheexample,wefindthatthispreservationofgeneral shape isnot true at the origin.This can be seenby referring to Fig.3-5 and observing that the rectangle B formed around the origin by the lines u=± 0.5,w=± 0.5transformsintoafigureB'whichhasnoresem- blance to arectangle. Thus,experiencegainedfromtheseexamplesimpliesacertaingeo- metrical regularity of the transformations, as embodied in preservation of anglesofintersection and approximate geometrical shapes,in going from one plane to the other.However, we also find certain exceptions.In the presentsectionweshallexplainthisbehavioroftransformationmaps, showingwhytheindicatedrelationshipsareusuallyfoundandunder what conditionsexceptionsaretobeexpected. Thederivativeservesasthepointofdeparture.Recallthatthe derivative ofw=f(8)at apoint 81isdefined as limAA w = 1'(81) .1...... 08 where As is an increment frompoint SIand Awis the corresponding incre- ment inthe wplane.Existenceofthis limit impliesthat corresponding to an arbitrary small E > 0there can be founda6 suchthat I~ :- I'(S1)I < E when IAsl< 6 and thereforelAw- I'(S1)Asl< EIAsl ThisinequalityisportrayedinFig.3-21,at(a)forII'(s1) I ~0andat (b)for I'(S1)=O.If we use the notationl'(s1)=Ae ia ,it is apparent from the geometry ofFig.3-21athat and ilAwl- AIAsli< EIAsl lang Aw- a- ang Asl< sin- 1 i (3-36a) (3-36b) 74COMPI,EXVARIABLESANDTHELAPLACETRANSFORM It is emphasized that relation(3-36b)istrue only if 1/'(81)1=A> E ThiscanbeunderstoodbyreferringtoFig.3-21b.Inthiscaseitis certainthat lieswithinthecircleshown,butnoestimateofthe angleof ispossiblemerelyfromknowingtheradiusandlocationof the circlebecauseWIisat the center of the circle. WenowallowEto approachzero,whilerecognizing that Edetermines 6andthat < 6.Clearly,wearedealingwithincrementswhich may be required in certain cases to bequite small,depending on the size ofA.However,solongas 1'(81)isnotzero,asEapproacheszerothe (a)r'(sIJ+O WI o FIG.3-21.TraIl8formationofanincrement,at apointwherethederivativeexists. conditionE1='II"and4>2='11"/2.Wemustthenuse3'11"/2(not-'11"/2)forthe angle of 8182.Otherwise the law of adding logarithms would not be valid. Asausefulby-productofthisdevelopmentwearenowreadyto consider I c ~ where C is a simple closed curve encircling the origin in a counterclockwise direction.The interpretation given aboveallowsustowrite {d8={31~=log 81- log8 ~ }c8}.1'Z where 8 ~and 81are two points directly over one another,but on adjacent sheets of the Riemann surface.Since C is counterclockwise,the angle of 81is 2'11"greater than the angleof8;.Thus Another important case is !c d8'2 - = J'II" C8 {d8 }c 8" (4-23) where C is the same curve we have been considering and n is apositive or negative integer not equalto1.There isno loss in generality if C is the circle 8=pe;. Then, if weusethe specificvalues8;=p/ -'II",81=p/'II",weget !c d8J.'1 d8f" --;;=--;;=ipl-"ej(I-"l. d4> c8I.'8-r =jp 1 -" f ~ "[cos(1- n)4>+ jsin(1- n),p]d,p INTEGRATION105 whichiszerowhenn¢1andconfirms Eq.(4-23)when n=1.Thus, (d8=0n¢1(4-24) j08" 4:-10.Cauchy Integral Formulas.Let asimple closed contour C lie in asimply connected regionof regularityofafunction f(s) ,and let 8bea pointinsideC.Also,letC 1 bea circleofradiusrandcenterat8, lying wholly within C. The function fez) z - 8 isaregularfunctionofz,ina doublyconnectedregion,witha neighborhood of s deleted, as shown inFig.4-17.Now,bythe princi- plesestablishedinSec.4-7,itcan be said that 0 c \91 z plane (fez)dz=(fez)dz(4-25) joz-sjo,z-s FIG.4-17. Substitution of path of integra- tionby acircleapproachingzeroradius. and then (fez)dz=(f(8)+ fez)- f(8)dz jo. z- sjo.z- s =f(s)(~- r f(s)- fez)dz j 0,z- sj 0,z- 8 In the firstintegral on the right repla(:ez- 8by 8'and dzby ds'to give an integral likeEq.(4-23),and thus f(s)(~= j27rf(s) jo,z- 8 To dealwith the second integral,note that the only possiblepoint where theintegrandmightbecomeinfiniteordiscontinuousisatz=s;but lim f(8)- fez)=!,(s) %-+,8- Z wbich isnoninfinitebecause fez)isregular at z=8.Thuswecan write I f(S)- fez)I< I!'(s) I + E=M Z-8 106COMPLEXV ARlABLESANDTHELAPLACETRANSFORM when z ison C 1,and so I j f(8)- fez)dz I< 2rriM c.z - 8 whererl istheradiusofC 1 •Furthermore,the integralisunchanged if rl approaches zero.In sodoing Mremains constant.The conclusion is that and so,finally, j f(8)- fez)dz=0 c.z - 8 f(8)= ~ .r fez)dz 21fJ}cz - 8 (4-26) This isthe Cauchy integral formulafor f(8).Equation(4-26)is not to be construed as a formula for calculating f(8).It is useful as a repre8enta- tionforf(8),tobeusedinlateranalyticalwork.Itsusefulnessstems partly fromthe fact that it includes an integral and therefore can be used to evaluate certain kindsofintegrals. If welookat Eq.(4-26)and formallydifferentiateundertheintegral sign,weget 1'(8)= ~ .rfez)dz 21fJ}c(z - 8)2 1"(8)=~jfez)dz 21fJc(z- 8)3 / 0there exists anumber Nsuch that when nm> N.However,in view of the inequality rule forabsolute valuesof sums and relation(5-33),it followsthat """ II gle(S)I IIgle(s) II Mle< E 1:-...1:-...1:-... when n, m>N, forall sin R.However,this isthe same as whichistheCauchyprincipleofconvergence.Convergenceisuniform because Nis independent of s when s is in R.Theorem 5-4provides the WeierstrassMtestforuniform convergence. NowreturntotheseriesofEq.(5-31),havingaknownradiusof convergence Ro.Wewrite Is- sol < Ro and defineMle= .... giving I Mle= .1:-01:-0 Note that the series on the right isknownto convergebecause < R o • Also,if and so,by Theorem 5-4, Is- sol lalells- sollelIfk .. lalc(s- So)le 1:-0 converges uniformly in the region \s- sol < Ro (5-36) 128COMPLEXVARIABLESANDTHELAPLACETRANSFORM 5-6.Properties of Power Series.As an immediate consequence of the uniformconvergenceinregion(5-36)wecan say fromTheorem 5-3that .. /ef(s)ds=l/e a.\;(s- so).\;ds(5-37) ,\;=0 where C is any curve offinitelength inside region(5-36).Furthermore, if Cisaclosed curve,each integral /e ak(s- SO)kds=0(5-38) and so/ef(s)ds=0 and from Morera's theorem it is concluded that f(s)isregular in region(5-36). Now let s be apoint in region (5-36) and C a circle also lying in the region, as shown inFig.5-3.FromtheCauchyintegral formula forthe derivative, df(s)=-.!..(f(z)dz(5-39) ds27f'jJe (z- S)2 and the integrand can bewritten .. FIG.5-3.Regionof uniformcon- vergence of series(5-40). fez)=ak(z- SO)k(5-40) (z- S)2(z- S)2 k- TheseriesinEq.(5-40)convergesuniformlyinthedoublyconnected closedregion shown shaded in Fig.5-3.This istrue because I ak(z- so)k I (z- s)2- T02 .. andlakl To k-O converges.Therefore,wecanuseTheorem5-3(notethat the theorem doesnot require C to bein asimply connected region)to obtain .. _1_(fez)dz={akeZ- so).\;dz 27f'jJc (z- S)227f'jJc(z- s)2 k-O Eachtermunderthe summationisthederivativeofak(s- so)\by the INFINITESERIES129 Cauchy integral formulas,and so (5-41) where 8is in the open region 18- 801< R ~ Note that this region cannot be closed with an equality sign,because if 8 wereto beon the boundary circle 18- 801=R ~ it would be impossible for C to be in the shaded region.Of course, R ~can be arbitrarily close to R o ,the radius ofconvergence,and sowecan state the followingtheorem: Theorem5-5.Afunctionrepresentedbyapowerseriesexpanded about a point So,with radius of convergence R o , is regular forIs- sol< Ro andthereforepossessesallderivativesforIs- sol< Ro.Furthermore, the nth derivative of f(s)is given by the series obtained by term-by-term differentiation of the original series n times; and the radius of convergence ofthe series forthe derivative isalsoRo. 6-7.TaylorSeries.Intheprevioussectionitwasestablishedthat, within its circle ofconvergence, apower series defines afunction which is regular at eachpoint whereit isdefined.Wealsoemphasizedthat the seriesisspecifiedfirstandthatthe series defines the function.Now we approachtheconversesituation, wheref(s)isspecifiedinsomeform other than the series Wearetodeterminewhetherthis formcanbeanexpressionforthe functionoveranypartofthecom- plexplane.Thewayto proceedis to seewhether the coefficients a" can c 8plane and z plane FIG.5-4.Contour of integration used to develop the Taylor series,and region of convergence of that series. be determined from the givenf(s) and whether the series converges tof(s). Assumethat f(s)isanalytic,selectanypointwherethefunctionis regular,anddesignatethispointas80.In general,f(s)willhavesome singularpointsSI,S2,etc.,but,inviewofthedefinitionofregularity, 130COMPLEXVARIABLESANDTHELAPLACETRANSFORM there must be aneighborhood of 80in which there are no singular points. Let Robethedistancefrom80toitsnearestsingularpoint.Referring to Fig.5-4,let Cbe acirclecentered at 80,with radiusR< Ro.Then, in the region 18- 801< R the Cauchy integral formula f(8)=-.!.(fez)dz 211".1Jez- 8 is avalid representation for f(8).The next step is to write 1111 1,---8=1,- 80+80- 8- 1,- 801- (8- 80)/(1,- 80) (5-42) Althoughwehave stipulated18- 801< Rasthe rangeof8,nowchoose anumberR'< Rand restrict 8to the region 18- 801~R'< R IntheintegralofEq.(5-42)1,isconfinedtothecircle11,- 801= R, and so / 8- 80I ~R'< 1 1,- 80R In Sec.5-3it isshownthat 1 -- =1+8+8 2 + 1 - 8 181< 1 and thereforein this serieswecan replace8 by(8- 80)/(1,- 80)togive 1=1+8- 80+(8 - 80)2+... 1- (8- 80)/(1,- 80)1,- 801,- 80 when1(8- 80)/(1,- 80)1< 1.Furthermore,sinceR' /R< 1,weknow that R'(R')2 1+][+][+... convergesandthus(R' /R)"can serveasMlcintheWeierstrassMtest, proving that the above series converges uniformly for I ~ I ~ R ' 1,- 80R and therefore for 18- 801~R' INFINITESERIES131 whenIz- sol=R.Thislatter condition issatisfied forthe integral in Eq.(5-42),and so we can replace the integrand of Eq.(5-42)by the series .. fez)=f(z)~(s- SO)k Z- S~(z- SO)k+l k-O and then perform aterm-by-term integration,withthe result .. - 12:- kffez) f(s)- 2-'(sso)()k+ldz 7rJCZ- So Is- sol~R' 1:-0 Thisisaseriesinpowersof(s- so)\wheretheintegralfactorsare constants.R' can be as closeas welike to R o ,and sothe radius of con- vergenceofthis series isRo.Finally,this result isconveniently written co f(s)=I ak(S- so).\:18- 801< Ro ,\;=0 (5-43) where 1jfez)d ak= 27rjC (z_SO)k+lZ (5-44) Certainlythesecoefficientsexist,sincetheintegrandisregularonthe path ofintegration. It istobeobservedthatEq.(5-44)isverysimilartotheCauchy integralformulaforthekthderivative,differingonlyintheabsenceof the factor k!.Accordingly,we can write (5-45) whichisidenticalinformtotheusualformulaforthecoefficientsofa Taylorseriesinrealvariables.Accordingly,theseriesexpansion giveninEq.(5-43)iscalledtheTaylor-seriesexpansionoff(s)about point so. In thederivation leadingtoEq.(5-44)wedesignatedCasacircleof radiusR.However,Eq.(5-44)isinvariantif Cisdistortedintoany simpleclosedcurveinsidethecircleofradiusRbutstillenclosingso. Thus, in Eq.(5-44) we arrive at the final interpretation of C as any simple closed curve enclosing point Sobut not large enough to enclose any points where f(s)is singular. ByvirtueofthisproofwehaveshownthattheseriesinEq.(5-43) converges in the regionIs- sol< Ro and, furthermore,that it converges to the original function f(s).The seriesnowbecomes anew representa- tionforf(s),validinthecircleofconvergence.Thisdevelopmentis COMPLEXV ARIABLBSANDTHELAPLACETRANSFORM important because it shows that forany analytic functionapower-series expansion is possibleabout any point wherethe functionisregular. The information provided about the radius of convergence is especially important.IntheprocessofarrivingatEq.(5-43)it wasestablished that R o , the radius of convergence of the series,is the distance fromBoto the singular point closest to Bo.Thus,ifthe locations of singular points of /(B)are known,it is immediately known fromsimplegeometry in the complexplanewhatwillbetheradiusofconvergenceforaTaylor expansion about anypoint;thereisnoneedtocarryout aconvergence test. As you develop an understanding of the implications of Eqs.(5-43)and (5-44),it isespecially important tounderstand that Eq.(5-43)is anew representation of /(B),differing fromthe originalrepresentationwhich is used for /(z) in Eq.(5-44).The original representation can be a"closed- form"representation[likesinB,B/(B+ 1),etc.],orit canbeaseriesin powersofB - where issomepoint otherthanBo.However,in the lattercaseBomustlieintheregionofconvergenceofthegivenseries. SincetheregionofvalidityofEq.(5-43)isgenerallydifferentfromthe region of validity of the original representation, it isvery important that theregionofvalidityshallalwaysbestipulated aspartoftheformula, as shown in Eq.(5-43). Intheabovestatementofpossibleoriginalrepresentations,the possibility ,that /(B)may originally be represented by a series in powers of B - Bowas omitted, in anticipation of aspecial consideration of this case. Suppose that /(B)is definedby aconvergent series ... J(B)=I- Bo)$ 1:-0 havingafiniteradiusofconvergence.Thisfunctionisacandidatefor representation by Eq.(5-43),and accordingly weseek the akcoefficients, ... a",=rJ(z)dz=\'r(z- Bo)"dz(5-46) 27rj} C(z- Bo)k+l27rj) c(z- Bo)k+l n,:,O whereC is asmall circle centered at Bowithin the regionofconvergence. The interchangeofintegration and summation operations isjustified by Theorem 5-3.Furthermore, fromEq.(4-24)it isknownthat !c (z-,,+I{ :: Therefore,Eq.(5-46)yields INFINITESERIES133 This result may seem obvious and trivial, but it expresses the important principlethatthereisonlyonepower-seriesexpansionaboutagiven point whichconvergesto agivenfunction,andthis isthe Taylor series. ThisfactisimportantbecauseEqs.(5-44)and(5-45)donot,inmost practicalcases,offerthesimplestprocedureforfindingthecoefficients. If some other procedure can be foundto give aseries whichconverges to therequired function,thisseriesmust beidenticalwithEq.(5-43).In thisstatementthereisnodeprecationofEqs.(5-44)and(5-45).On manyoccasionstheyareindispensablebecauseoftheirgenerality, particularly in the subsequent proofs ofgeneraltheorems. Asan illustrationofthe convenienceofusingalternativemethodsof obtaining aTaylor series,consider the function 11 f(8)=(8- 1)(8- 2)8 2 - 38+2 expanded about the point 80= O.Wecan perform adivision algorithm as follows: +%8+ %8 2 2- 38+8 2 11 1 - %8+ %8- %8- %8 2 +%8 3 %8 2 - %8 3 %8 2 - 2H8 3 +%8 4 1%8 3 - %8 4 For the finitenumber ofsteps shown, f(8)=+%8+%8 2 + wherethequantityinparenthesesistheremainderterm.Without carryingoutthedetails,itisapparentthat forsmall181the remainder approaches zero as the algorithm iscontinued; the series +%8+%8 2 +lX683+... convergestof(8)for1812 88 2 8 3 8 4 For the Taylor series(region1)weneed181< 1 and therefore weadd the firstseries of each of these sets to get 181 AN'+ --- y2'11"E By wayofthenumbering onthesequenceofcurvesC 1 ,C 2 , willestablish the number Nsuch that .!.I( ~ M d z l 0 164COMPLEXVARIABLESANDTHELAPLACETRANSFORM 1-11.By followingthe pattern (. (1+ z)dz=- (1+ 8)2_! Jo22 ate.,derive the binomial formula(for integer n> 0) .. (1+ .)w- l kl(n ~k)I'· 1:-0 1-12.Starting withthe power series for1/(1+ 8),obtain aseries forlog(1+ 8). Justify your steps,and state the regionofconvergenceof the newseries,explaining howthis region of convergence was determined. 1-18.Given the series 2 1 8 1 2'8' c082,-1- 2f + 4f + obtain series expansions for (a)Sin l 8(b)COSl8 (HINT:Consider term-by-term differentiation.) 1-14.Given the power series .8'8' SIn8 "*8-3i+5i 8 1 8' COS8=-1- 2i + 4i ... (c)sin 8cos8 (a)Determine the radius of convergence of each series. (b)Starting withtheseseries,andwithoutsquaringthecosineseriesorwithout usingthederivativef6rmulasfortheTaylorcoefficients,obtainthreeterms ofthe power series in 8for 1 and findits radius of convergence by any justifiable method. 1-11.DoProb.5-14 by evaluating thefirstthree coefficientsof theTaylor series. 1-16.By any method obtain series expansions, in powers of If+ 1, for the following: 1 (b)1+ ,. 1 (d)41f- Ifl In each case specify the radius of convergence.. 1-17.UsetheCauchyintegralformula forthe nth derivative ofafunction 1(8)to provethat foraTaylor series Of I(a)=l aw(,- 8,)w .. -0 the coefficients obey the inequality whereMr=- max1/(8)1on18- 8,1- T< [radius of convergence). 6-18.Givenapolynomial INFINITESERIES N /(3)=l a,.3" o Useappropriate ideas relating to series to show that it can be written N where 6-19.Given aLaurent series /(8)=L A.{s- 1)· 1:-0 N A"nla 3 •=~kl{n- k)! .. -I: 165 whichconvergesforR,< \8\< R t •Determineitsregionofuniform convergence. 6-20.For the function 8 2 + 8+ 3 8·+ 28'+ 8+ 2 obtain the followingexpansions,and in each case establish the region of convergence: (a)Taylor expansionabout 8=0 (b)Taylor expansion about s=-1 (c)Laurent expansions(twoof them)about 8=0 (d)Laurent expansion about each singular point 6-21.Carry out the tasks specified in Prob. 5-20,but for the reciprocal of the func- tion specified in that problem. 6-22.Youare given the function 1 /(8)={28+ 1)(8_l)t For each ofthe followingcases specifythe region(orregions,if morethan one series ispossible)ofTaylorand/orLaurentexpansions.ThecasesareforexpanSIOns about the followingpoints: (a)8=0 (c)8=- ~ 6-23.The function (b)8=1 (d)8=-2 1 sin(1/8) issingular at the origin.Showthat the Laurent expansion about the origin forthis functionhas zeroradius of convergence(i.e.,such an expansion doesnot exist). 6-24.Obtain the appropriate series expansionfor (a)Expanded about 8=0 (c)Expanded about If=2 /(8) .,.. z:sm,_l (b)Expanded about 8=1 166COMPLEXVARIABLESANDTHELAPLACETRANSFORM 1i-21i.For eachofthe followingfunctions,locatethe singular points,andidentify whethertheyarepoles(andofwhatorder)oressentialsingularities(andofwhat kind): (a)~ 8 (b)ell' (c)6- 11 , 8 2 (d)(82 + 1)2 (e)sin 8 1 (f)sinh II 1i-26.For each of the followingspecifywhether the functionis regular or singular at infinity, and if it is singular, specify whether it is apole or an essential singularity and,ifthelatter,what kind: 6' (a)1- sin 8 (c)6"- 6' 8 3 - 28 2 + 8 (e)88+ 38" 8 8 - 2 (b)s"+ 2 (d)tan 8 (f)sin 8 , 1i-27.Find the residues at the indicated singular points forthe following: (a)sin 8 8' 1- e- 2I (c)-8-'- e 21 (e)(8- 1)2 at 8=0 at 8=0 at 8=1 6-28.Given the function 1 (b)81_8" ( d)cos II sin 2 8 tan II (f)(1- 6')2 1 /(8)=IIsin II (a)Locate and classify its singularities. (b)Evaluate the residues at these singular points. at 8=1 at II=-,.. at 8=0 (c)Evaluate the integral of /(8)in acounterclockwisedirectionaround acircle of radius 5,centered at the origin. 1i-29.(a)Use the method of residues to evaluate the integral (8dB Jc 1- e' whereC is the rectangular path shown in Fig.P5-29. (b)Let 10 designate the answer to part a.In terms of 1 0 , what is the above integral if the contour is changed(1)toC 1 and(2)toC.? 88 C -8 8 -88-44 -8-8 FIG.P5-29 INFINITESERIES167 6-30.Evaluate the integral (8' + 1 Je (8- 1)(8"+ 4) d8 whereC is acounterclockwisecircle centered at 2 and of(a)radius 2;(b)radius 4. 6-31.Let C be the unit circle,with counterclockwise sense of integration.Evalu- ate each of the following: (b)fe 8s': 8 6-32.Evaluate the integral over each of the followingcounterclockwise paths: (a)C A ,aunit circle 'centered at 8=0 (b)C b ,aunit circle centered at 8= i (c)C.,acircle ofradius 2 centered at 8...0 6-33.LetC designateasemicirculararcofradiusR,centeredatasimplepole 80 and subtending an angleBo.LetAobethe residue at the pole.Prove that lim(/(8) d8= jBoA. R_oJe 6-34.Youarep;iventhe series Determineitsradiusofconvergence,andobtaina"global"representation.Also, obtainaseriesfortheelementofthisfunctionexpendedaboutpoint8=_ %. 6-36.Obtain aglobal formula forthe functiondefined by the series 6-36.Show that the two series .. ~2Hl-3 /1(8)=~( - 2 ) ~(8- l)t .1:-0 .. and /.(8)=L[( -~ r-2 (- ~ r ](8- 2)k .1:-0 are elements of the same analytic function.(For ahint, seeProb. 5-10.) 6-31.Obtain the partial-fraction expansion of the functiongiven in Prob.5-22. 6-38.Obtain the partial-fraction expansion of the function given in Prob.5-20. 6-39.Obtain the partial-fraction expansionof the reciprocal of the functiongiven in Prob. 5-20. 168COMPLEXVARIABLESANDTHELAPLACETRANSFORM 6-40.Obtain two terms of the Taylor expansion for res),in powers of s, for Example 2 in Sec.5-15. 1-41.Obtain the partial-fractionexpansion of sins 8(8- 1)(8- 2)2(8- 3)1 including two terms in the Taylor expansionof res),in powers of s. 5-42.Obtain the partial-fraction expansionof /(s)=cosa (8- 1f'/2)(s- ,..)(s- 2... ) including two terms of theTaylor expansionofres),in powersof 8. 5-43.Derivethe formula .. tan s=I n-! (odd) 1-44.Derive the formula .. cots=! s "\'12s nOr"1- (slnr)" n-! 1-45.Obtain theMittag-Leffler expansionfor sins /(s)=cos2s 6-46.From the formula given in Prob.5-44derive the infinite-product representa- tion .. sins=sn (1 s") - nOr' n=1 [HINT:Observe that cot 8=d(logsin s)lds.] 5-47.From the formula given inProb.5-43derive the infinite-product representa- tion coss=n (1 n=l (odd) [HINT:Observe that tan s=-d(log coss)lds.) 6-48.Supposeananalyticfunction/(s)=p(s)lq(s)hasaremovablesingularity at apoint 80due to pCs)and q(s),each having azeroof order nat 80.Prove that /(s) approaches the limit p(ft)(s)I lim/(s)=-- '-"q(ft)(s)1_', Note that this has the appearance of Lhopital's rule applied to a function of a complex varia.ble. CHAPTER6 MULTIVALUEDFUNCTIONS 6-1.Introduction.Having established the concept ofasingle-valued function,W = f(8),wenownaturallyaskwhethersuchafunctioncan always have an inverse whereby 8can be specified as a function ofw.In those caseswhereseveral values of 8yield identical values of wweare in trouble,forthentheinversecannotbesingle-valued,andinthetrue senseofthewordan inversefunctiondoesnot exist.The maintask of this chapter is to developa method ofanalysis which willpermit" multi- valuedfunctions"tobetreatedatleastpartiallylikesingle-valued functions. Wecandrawsomeexamplesfromtherealmofrealvariables.The function y=sinx issingle-valued,but the inverse x=sin- 1 y ismultivalued,asillustratedinFig.6-1a.The samecommentscanbe madeabout y=x 2 and its inverse x=yy, which is shown in Fig.6-1b.Probably your experiences with the square- root.function,andtheproblemofchoosingsignsinthecaseofreal variables,has pointed upthe need forisolating thesecases. In eachoftheabovetwoexamplesagivenfunctionissingle-valued, andits in'Terseismultivalued.There areothercaseswhicharemulti- valued"both ways."An example is y2 -=x 2 -1 which is showngraphically in Fig.6-1c. Whendealingwithcomplexvariableswesometimesfindmulti- valuednesr- whichdoesnot appear in the real-variable counterpart.For example,illChap.4wemet themultivalued functionlog8.However, 169 170COMPLEXVARIABLESANDTHELAPLACETRANSFORM log x(where x> 0 and real) is not multivalued.Thus it is apparent that graphical illustrations like those of Fig.6-1are inadequate for the general case of a function of a complex variable.It is hoped that ultimately you willconclude that multivalued functions are simpler to understand when thevariableiscomplexthan whenit isreal.This simplificationcomes about through the concept of a Riemann surface.You met this briefly in thediscussionoflog8inChap.4andwillseemuchmoreofit inthis chapter. ---=oo!-""'+---y- ~ - + - ~ ~ - - - - - - y - - - - - 1 - ~ - - - - y FIG.6-1.Examples of multivalued functionsof areal variable. 6-2.Examples of Inverse Functions Which Are Multivalued.Perhaps the simplest multivalued functionis the inverseof W=8 2 (6-1) which will bewritten 8=w ~ Theexponent% intheaboveequationisdefinedasanotationwhich impliesthe inverse ofEq.(6-1). The necessaryideas forstudying the inverseofEq.(6-1)wereantici- patedinFigs.3-3and3-4.If thetwowplanesofthosefiguresare regardedasbeingidentical,areasA' and B' are identical and the above functionalrelationshipwouldcarrythisareaintoareaAorareaBof the8plane.From theformulaalonetherewouldbenowaytodiffer- entiate between areasAand B. Wecouldcontinuetoregardwastheindependentvariablewhen analyzingtheinverseofthosefunctionswhichhavepreviouslybeen considered.There wouldbesomeadvantage indoingthis,particularly inconsideringmappingproperties,becausethenlabelsontheplanes wouldremainunchanged.However,thereareadvantagesinalways using8astheindependentvariable;andsinceweshallbeconsidering functionsotherthantheinversesofpreviouslytreatedsingle-valued functions,weshall continue to use8as the independent variable. MULTIVALUEDFUNCTIONS171 Accordingly,theinverseofEq.(6-1)isnowwrittenwith8andw interchanged,as follows: w=8 ~ (6-2) • This function isdescribed by Figs.3-3 and 3-4if thew- and s-plane label8 are interchanged,sothat now there will be two splanes which map onto a singlewplane. Weshallnowexploitthe ideaofhavingtwo8planes.If somehowa distinction can be made between these two s planes, wecould then regard overlyingpointsinthetwoplanesasbeingdifferent,andthefunction w=s ~would appear to be single-valued.To dothis necessitates over- cominganobstacleintroducedbythewedge-shapedcutsalongthe negative real axis.The difficulty issurmounted by the ingenious device of imagining the two planes to be attached along the cut edges.Referring to Fig.6-2,foreach pair of edges consisting of one edge from each plane, onesolidlineand onedashed linefittogether.Thencurves suchas0 and0' donot crossacutbutpasscontinuouslyfromoneplanetothe other.Whenthetwosplanesarejoinedinthisway,theyforma Riemann surface.Each ofthesplanes iscalled asheetoftheRiemann surface;and the cut in eachsheet iscalledabranchcut.Apoint likeS6 in Fig. 6-2 is called a branch point.That portion of the function described whensisinonesheet iscalled abranchofthe function. Suppose that there are two points S1and s ~similarly located in the two sheetsofFig.6-2.TheRiemann-surfaceinterpretationallowsthemto beregardedasdifferentpoints.Inthisway W1= f(s1)and w ~= f(sD are clearly distinct because ang s ~=211"+ ang S1.With this interpreta- tionf(s)becomessingle-valued.Manytheoremsoriginallyprovedfor single-valuedfunctionsnowbecomeapplicablein the multivalued case. Sinceit isimportanttobeabletoidentifyapointwithaparticular sheet,it isnecessaryto have amethod ofkeeping track ofthis.Wedo soby consideringtheangularpositionofalinedrawn fromthebranch pointtothepointinquestion. tIn thecaseofFig.6-2thisismerely theangleofthevariables.In sheet1thisangle(q,)liesintherange -11"< q,;;;!11",and in sheet 2the rangeis 11"< q,;;;!311". In order furthertoexplaintheseconcepts,considerneighborhoodsof pointssand s',wheretheunprimedvalueisalwaysinsheet1andthe •Thenotationw=Vaispurposelyavoided.InthischaptertheVsymbol willbereservedforusewithpositiverealnumbers,andwhenthesymbolisused,a positivesignwillbeunderstood.InChap.10the symbolV8isused,butwitha specificmeaning definedthere. t Polarcoordinatescenteredattheorigincanbeusedtoidentifywhichsheeta point isin onlywhen there isabranch point at the origin.When abranch point is at some other point, as in some of the later examples, an auxiliary polar-coordinate sys- temiscentered at the branch point in order to accomplishthis task. 172COMPLEXVARIABLESANDTHELAPLACETRANSFORM primedoneisinsheet2.Eachoftheseneighborhoodswillbetrans- formedintoneighborhoodsofcorrespondingpointsinthewplane.A fewparticularcasesareconsidered,beginningwithpoints81and8 ~ . There isnopossibilityofthe neighborhood of 8 ~becoming confused with theneighborhoodof81.Thispermitsustousethedefinitionofcon- tinuity without being bothered by multivaluedness. if:II Areaof mapAreaof map - of sheet 2of sheet 1 FIG.6-2.Riemann-surface interpretation of the functionw...B ~ . Apoint like82ona"solid-line"edgeofabranchcut cannothavea neighborhoodwhollyinonesheet.Itsneighborhoodmustbeintwo sheets, as indicated by the two shaded areas in Fig. 6-2.This neighbor- hood goes into aneighborhood of W2in the w plane.The corresponding point8 ~hasaneighborhoodconsistingofthetwononshadedcircular segments,whichtransformsintoaneighborhoodofw;.Althoughthe neighborhoodsof82and 8 ~are each in twosheets,the functionissingle- valued in eachneighborhood. * Wenowcometotheuniquefeatureofabranchpoint.If wetry to put a small circle around s" in sheet 1,wefind that points a and b cannot • Later on it isshown that choiceofthe branchcut is arbitrary.For adifferent choice,say along thepositive realaxis,Bland B;wouldeachhaveaneighborhoodin asingle sheet. MULTIVALUEDFUNCTIONS173 be connected; from point a we must proceed into sheet 2.If points a and bareallowedtoapproacheachother,thecorrespondingpointsinthe w plane approach a' and b', which are at the ends of a semicircle, as shown inFig.6-2c.Asmallcirclewhichencirclesabranchpointonlyonce cannottransformintoaclosedfigureinthefunctionplane.Twoor more circuits (two in this example)around abranch point are required to give aclosed figurein the functionplane.Branch points are designated by an ordernumber.Theorder isonele88than thenumber ofcircuits around it required to give aclosed figurein the functionplane. Theabovedescriptionbringstolightotherdistinctivefeaturesofa branch point.Unlike points such as 81and 8z,abranch point cannot be assigned to anyone sheet of the Riemann surface, and therefore it cannot haveaneighborhoodlyinginonlyonesheet.That is,it isimpossible todefinea.neighborhoodofabranchpointinwhichthefunctionis single-valued. The factthat encirclingabranchpointonlyoncedoesnotclosethe figuretraced inthe functionplanecanbeusedto test whetheror not a givenpointisabranchpoint.Asanexample,weshalltestwhether 8=0and 8=1 arebranchpoints ofthe function At 8=0wewrite giving and w= 8j.S 8= pi'; w=rei' r=VP If tPis increased by 211",so that point 8= 0 is encircled once, 8 will increase by 11",which willcarry w only halfway around the origin.Thus,8= 0 is abranchpoint.Nowlookat thepair ofpoints8=1and w=1.In their neighborhoods wewrite 8=1+ pe i ';w=1 + re i9 and1 + 2re i '+ r Z e i26 =1 + pe i '; As r is made very small,the rZterm approaches zero faster than rand 80 the aboveapproaches showingtha.tpoint w= 1 isencircledonlyonce when8= 1 isencircled onceby asmallcircle.Thus,8=1 is not abranch point. Wehaveseenthat the functiondescribedby Eq.(6-2)hasabranch point at 8=O.If theRiemann-sphereinterpretation is introduced,we can alsoidentify abranchpointat thepoint infinity.Asmallcircular pathenclosingthepointat infinityontheRiemannspherebecomesa 174COMPLEXVARIABLESANDTHELAPLACETRANSFORM large circle in the flat plane.Thus, to test whether the point at infinity is abranch point, we look at the figuretraced in the function plane as we follow one circuit around a large circle (approaching infinite radius) in the 8 plane.If the function-planefiguredoes not close,the point at infinity isabranchpoint.Thepointatinfinitycanalsobeinvestigatedby exaIniningf(I/8)at the origin. Thusit isconcludedthatthefunctionw=8J.ihasbranchpointsof order 1,at 8=0 and at infinity.They are located at ends of the branch cut.Everymultivaluedfunctionhasabranchpoint at eachendofa branchcut.Asweshallseelater,somefunctionshavebranchcuts extending between pairs offinitebranch points. Wecanlearnabitmoreaboutinversefunctionsbyconsideringthe inverse of 8=w· (6-3) which is conventionally written (6-4) wheretheexponent% isdefinedtomeantheinverseofEq.(6-3).In this casethe Riemann surface has three sheets,each ofwhich maps onto one-thirdofthewplane.Withalittlethoughtyouwillseethatit is necessary to encirclethepoint 8= 0three times in order to get aclosed figurein the w plane.Also,it is evidentthat infinity isabranch point and that both branch points are of order 2.The interconnection ofedges ofbranchcutsisillustratedinFig.6-3by curvesC,C ' , andC"and by thesequenceofnumbers.Points2and3,4and5,and6and1are, respectively,connected together. Asafinalexampleinthissectionconsideramultivaluedfunction having an inversewhich isalsomultivalued.The casein point is (6-5) which can be written w 2 =(8- 1)(8+ 1)in order to show that, if either point8 = 1or 8=-1 isencircledtwice,thenpointw= 0isencircled once.Thus,points-1and+ 1arefirst-orderbranchpointsinthe 8plane.Thesearethebranchpointsofwasafunctionof8.Toget the branch points in the w plane,for8asafunctionofw,wewrite 8 2 =(w+ j) (w- j) whichshowsthat in the w plane therearebranchpoints at jand-j. The complete representation of this function requires Riemann surfaces oftwosheetsforeachvariable,as showninFig.6-4.Branchcuts are indicated by the double lines.This situation is too complicated to adInit acomplete graphical picture.We shall consider only the transformation fromBtow.BranchpointsatB=+ 1and-1 areenclosedbyfour MULTIVALUEDFUNCTIONS175 circles, which go into the four semicircles with similar labels in the w plane. A fewrectangular-coordinate linesin the 8plane and their traces in the w plane are alsoshown. The distortion ofthe 8plane in going to the w plane can be visualized by thinking ofthe two sidesofthe branch cut being pulled apart in the direction of the arrows,while the branch points move together.Further interpretation of the mapping ofthis function isobtained by considering w-plane Rangefor sheet 1 FIG.6-3.Riemann-surface interpretation ofthe functionw=B ~ . whathappensingoingfromonesheettotheotheratthebranchcut. For example,whengoing fromAtoA' in the 8planethe corresponding w point goes along the line with similar labels in the w plane.At A Ithe branchcut isencountered inthe 8plane,and acontinuation ofthis line must be B'B in the other sheet.The corresponding line is also shown in thewplane.Thus,ingoingalongaverticallineinthe8planewhich "crosses"abranchcut(notactuallycrossing,buttransferringtothe other plane)we findthat we go to the w-plane branch point at w=jand transfertheretotheothersurfaceinthewplane.Thisisconsistent HI:)COMPLEXVARIABLESANDTHELAPLACETRANSFORM withtheinterpretationofw =+j and-j asbranchpointsinthe w plane.Lines ee' and D'D show what happens in amore general case. 6-S.The LogarithmicFunction.The function w=log 8(6-6) wasintroducedinSec.4-9.Anintroductorydiscussionofthereason fordefiningtheRiemannsurface,andaperspectiveportrayalofthe Riemannsurfaceforthelogarithmicfunction,isgiveninFig.4-15. o o o o ----t- o o o A Sheet1 s-planesurfaces I I o o I o o .--t-. -'[t"I BD"1" I Sheet2 BDI A -.- "--"- "Sheet1 " " " "w-planesurfaces B FIG.6-4.Riemann surfaces inthe w a.nd8pla.nesforthe functionWI=8 1 - 1. Also,thefunctionw=e'istreatedgraphicallyinChap.3;andif we write8=e W instead,wehave w=log8as itsinverse.Thus,Fig.3-13 describes the logarithmic function if the 8and wlabels are interchanged. The graphical interpretation ofEq.(6-6)is shown in Fig.6-5,which may profitably be compared with Figs.3-13and 3-14. I t is to be emphasized that there are an infinity of sheets in this Riemann surface.Each sheet maps ontoan infinite strip ofthe wplane.If the branch cut is along the negative real axis, the w-plane strips have edges at oddmUltiplesof11",as shown. MULTIVALUEDFUNCTIONS 177 ApathencirclingtheoriginintheRiemannsurfaceofthesplane producesaverticallineinthewplane.Thus,forafinitenumberof encirclements the path inthe w plane doesnot close.But by admitting the point at infinityin the w plane this vertical line may be said to close at infinity, after an infinite number ofencirclements.Thus,it is reason- able to designate the s-plane points at zeroand infinity as branch points of infinite order. Range of sheet 2 ' I I ~Rangeof" 1sheet! Iw-plane Rangeof sheet 3 FIG.6-5.Riemann-surface interpretation of the functionw=log8. 6-4.DifferentiabilityofMultivaluedFunctions.Werecallthatif a functionis to have aderivative it isnecessarythat ~ h elimit 1 ·f(s)- f(so) 1m'--'---'-----'-"'---'- ........s- So shallexist.Ingeneral,thislimitcannotexistiff(s)isnotasingle- valuedfunctionofs,becausethechoiceoff(s)wouldnotbeunique. However,letSobe definedtobeinonesheet ofaRiemann surface,like Sl in Fig. 6-2,and let s be in its neighborhood in that sheet.Under these conditions,f(s)- f(so)isauniquefunctionofs,andthelimitofthe differentialquotientcanexist.Whenitdoesexist,itiscalledthe derivative of the function.Now suppose that Socoincides with abranch point.Aspreviously shown,at abranch point it isimpossibleto define aneighborhoodinwhichthefunctionissingle-valued.Therefore,the differential quotient has no unique value, and no limit can be taken.On thebasisofthe aboveargumentsit isconcludedthat noderivativecan exist at abranch point.Thus,in addition topolesand essentialsingu- larities wehave athird class of singularities,the branchpoints. In consideringthequestionofdifferentiabilityat abranchpoint it is necessarytomentionthat acertainpointmaybeabranchpointwith respecttoseveralsheetsoftheRiemannsurfacebutmaybearegular point in other sheets.You willfindan exampleofthisphenomenon in Sec.6-8. 178COMPLEXVARIABLESANDTHELAPLACETRANSFORM As we discuss the general case,let an n-valued function have abranch point of order m at so.From the known properties of abranch point we can say that Somust lie in m+ 1 sheets.However,it ispossible,as in Fig.6-6,forntobegreaterthanm+ 1,inwhichcasetherewillbe n- m - 1 other sheets in which the corresponding point is not abranch point.The function may have a derivative at these other points.Thus, weneedto includethe abovequalificationwhenstating that afunction hasnoderivativeatabranchpoint.Thestatementistruewithout qualification only if the order of the branch point is exactly one less than the multiplicity of the function. It is important to observe,referring to Fig. 6-6, for example, that there mustbetwoormorebranchpointsinsheet3,becausethat sheet must share abranch cut with at least oneother sheet. \" Branchpoint ~ 8 0 Sheet1 jBranCh point ~ ISheet 2 Not a ~ b r a n C hpoint ~ -·1 Sheet3 FIG.6-6.Exampleofafirst-orderbranchpointforathree-valuedfunction.No derivative caq exist at 110in sheets1 and 2,but it canexistat 80in sheet 3. Two important conclusions are derived fromthe above discussion,one being that at points other than branchpointsthe derivative ofamulti- valued functionmight possiblybe multivalued,and the other that there can be noderivative at abranch point. Althoughwekeepinmindthedefinitionofthederivativeasalimit processinthevarioussheets,wecanusealreadyexistingknowledgeof thederivativeto obtainthederivativeinthevarioussheets.Allcases of interest can bewritten in the implicit form g(s)=hew)(6-7) whereg(s)and hew)are analytic functionsofsand w,respectively.At corresponding points sand wsatisfying Eq. '(6-7),and wherethe deriva- tivesg'(s)andh'(w)exist,it followsfromthe theory ofderivatives that giving The function dw g'(s)=h'(w)ds dwg'(s) ds=h'(w) (6-8) MULTIVALUEDFUNCTIONS 179 is an example of Eq. (6-7), and from it we get the derivative of w= s!oSby usingEq.(6-8),asfollows: dw11 ds=2w=2 s ~ or d ( s ~ )1 (f8 = 2sli This result is in agreement with the usual rules fordifferentiation.Note that the derivative is multivalued.Let tPbe in the range-'I/'< tP~'I/'so that in sheets1 and 2s isdesignated,respectively,by pe i + and pe M + 2r ). Then the derivative is,respectively, 1 2ype i +'2 and 11 2 V p ei o. If the function has finite discontinuities on these loci,there is no difficulty inshowing that the contributionof the S"rectanglesapproacheszero. In this brief discussionwehave omittedmany details.Inparticular, we did not take into consideration the special case where the locimay be eitherhorizontalorvertical.Thiseausesnodifficultybutdoesnotfit intothisanalysisbecauseitwouldgivekalimitingvalueof0or00. However,to adegreewehave establishedthe following: Theorem 8-7.If !(x,y) is almost piecewise continuous in x and yin the intervals axband cyd,except at possibly afinitenumber of points,then J. ddy !"b !(X,y) dx and both exist,andthey are equal. An example may behelpful.Let x !(X,y)= vl x _yl •This brief intuitive argument assumes that a single multiplying constant M in the above appraisalwillsufficeat allpoints.Otherwise,this argument isnot valid.In view gf the intuitive nature of this proof, it would be inconsistent to pursue thisques- tion indetail. THEOREMSONREALINTEGRALS 247 which is APe, with asingular point at x=y.The locus of the singular point isalineat 45°.Let us checkwhether or not (1(1xdx 10dy 10vlx - y\ and eexdy 10dx 10vlx - yl are identical.For the firstone, ~ = = =+=%y% 10 1xdxlollXdx11Xdx ovl x - yl0Vy - XIIV X - Y + % (1+ 2y)VI - y and,forthe secondone, (1xdy=('"xdy+(1xdy=2x%+ 2x VI _ x 10vl x - yl10V x- y) '"V y- x by ordinary processes of integration.Integrate the first from 0 to 1,with respecttoy,giving % Jo 1 y%dy+ % Jo 1 (1+ 2y).yr-=y dy =% Jo 1 y%dy + % Jo 1 (3- 2w)VW dw = % Jo 1 y%dy- % Jo 1 w ~ 'dw + 2 Jo 1 VW dw=2 Jo 1 VW dw=% wherew=1- Yisusedasavariablechange.Thesecondone,inte- grated with respectto x,gives 2 Jo 1 x ~ ·dx+ 2 Jo 1 xVI - xdx=2 Jo 1 x%dx+ 2 Jo 1 (1- w)VW dw =2 folx%dx- 2 folw%dw (1- (1_;- + 2 J0V wdw=2 J0vwdw=% where w=1 - x.Both results are the same, in agreement with Theorem 8-7. 8-7.Iterated Integrals ofFunctions of Two Variables (Infinite Limits). Asan extensionofthe previous section,supposethat I(y)=fo"f(x,y)dx(8-16) converges forsome range ofy,say c~y~d.The question is whether or not !cd dy 10" f(x,y)dx andfo"dx f d f(x,y)dy 248COMPLEXVARIABLESANDTHELAPLACETRANSFORM !!oreequal if f(x,y)meets the conditions given in Theorem 8-7,forevery finiterectangle.You may recognizeherearesemblance to the term-by- term integration of series,as discussed inChap.5. Supposethat,givenasmallarbitrary positivenumberfl,it ispossible to findanumber X,whichdependsonlyonE,suchthat II A ~f(x,y)dx I < fl whenA, A' >X for ally in the range c ~y~d.If this is possible,the integral is said to be uniformly convergent with respect to y,in the range specified.(This is the Cauchy versionofthe definitionofuniformconvergence.) Now assume that the integral with respect to xis uniformly convergent, and observethat the integral can bewritten as aseries .. 10" f(x,y)dx=lB .. (y) .. -0 (8-17) where h IJ .+. B .. (y)=f(x,y)dx IJ. (8-18) andthesequence0=(:Jo< {31< {32< (:Ja•••isanyinfiniteascending sequence of numbers starting from zero.Let such asequence be chosen, and chooseanumber Nsuchthat {3N>X Then,becausethe sequenceof {3'sis ascending,it istrue that {3.+h(3,.> X whenU,v> N Wearbitrarily let v bethe larger of the twon4mbers uand v,ifthey are not equal.Note alsothat • It+· f(x,y)dx=lB,,(y) n=u However,owing to uniform convergence ofthe integral and the appropri- ate choiceofuand v indicated above, II~ . + .f(x,y)dx I < I! whenu,v> N and consequently whenu,v > N THEOREMSONREALINTEGnALS 249 Thus,it is seen that uniform convergenceofthe original integral ensures uniform convergence of any series constructed from it in accordance with Eqs.(8-17)and (8-18).From Theorem 5-3,it is known that a uniformly convergent seriescan be integrated term by term,and sowehave .. fl dyfo"f(x,y)dx=feddyL B .. (y) ,,-0 .. = L fedB .. (y)dy n-O .. =lfeddyftt+1f(x,y)dx ,,=0 .. =L ft-' dxfe d f(x,y)dy .. -0 Wenowrecallthatthesequence{jo< (jl... isarbitrary,andunder thisconditionthe summation ofthe last expressionyields fo 00dxfed f(x,y)dy Thus,wehaveproved the followingtheorem: Theorem8-8.If f(x,y)meets the conditions given in Theorem 8-7, for every finiterectanglein the xy plane,and ifthe integral fo 00f(x,y)dx convergesuniformlywithrespecttotheupperlimit,Intherange c~y~d,then fe ddyfo 00f(x,y)dx=fo 00dx J. df(x,y)dy A similar theorem can be proved regarding inversion ofthe integration order of fedz fo 00f(x,z)dx wheref(x,z)isafunctionofarealvariablexandcomplexvariablez. However,there is aslight difference: almost piecewise continuity has not beendefined forafunctionofacomplexvariable,and so this isavoided by requiring f(x,z)to be continuous in zforallvaluesofxforwhich the functionisdefined.Theprooffollowsbyrecognizingthatacontour integralcanbewrittenasthesumoffourrealintegralsandthenby applying Theorem 8-8 to each ofthe realintegrals.The result isstated as follows: 250COMPLEXVARIABLESANDTHELAPLACETRANSFORM Theorem8-9.If fex,z)isAPe in xforfixedz and continuous in z for each xforwhichthe fuactionisdefined,and if fo"fex,z)dx converges uniformlywithrespecttotheupper limit,forz on acurve 0, then Iedzfo"fex,z)dx=fo"dxfeJex,z)dz 8-8.Limit under the Integral for Improper Integrals.In dealing with theLaplaceintegralweshallhaveoccasionalneedtoknowwhetheror not,givenafunctionoftwovariables J(x,z),wecanwrite lim(..fex,z)dx=('" limfex,z)dx .-20 )0Jo&'-Zo Inordertoappreciatethatthisisaquestionofinterest,considerthe integral r" sinydy }oY whichis evaluated inSec.8-12.Nowlet z> 0bercal,and lety= zx, giving r'"sinzx d_r }o-x- x- 2 This is afunctionofz havingthe limit z>o rr'"sin zx dx_r . ~ } o-x- - 2 However,ifwetakethelimit beforeintegrating,weget 10 .. I·sinzx d- 0 Im-- x- o. ~ ox showing that different result,Rare obtained depending on whether wctake the limit fin-itor integrate first.Thcrefore,conditions are wantcd which are sufficienttopermit an intcrchangeofthelimit andtheintegral. It is asimplematter to showthat theRclimits canbedifferentonlyat points wherethe integral is adiRcontinuous functionofz.Inthe above example,ifzapproacheszcrofromthenegativeside,theintegral approaches-7r/2,confirmingtheabovestatement forthisexample. The followingtheoremis adequate: Theorem8-10.Let afunction J(x,z)beofthe form J(x,z)=g(x,z)h(x) THEOREMSONREALINTEGRALS251 whereziscomplexandg(x,z)iscontinuousineachvariablewhenthe other variable is held constant and where h(x) is APO.If the integral fo"I(X,2)ck converges uniformly with respect to the infinite liInit,in aregionR, it is true that lim(ooI(x,z) ck=( ..lim I(x,z)ck ....... ,.)0}o,-tIt ifZoliesin R,and zlies in Ras it approaches zoo Weshallomittheproofofthistheoremwiththecommentthattwo considerations arenecessary.Sincethe integrand isAPe in variablex, it is necessary to consider the validity of the result in the light of singular points of the integrand at finite values of x, wherethe integral is improper. By restrictingbehaviorat thesepointstobeingoftheAPetypethese singularpoints cause nodifficulty.Then it isnecessarytoconsiderthe effect ofthe infinite limit, and in doing this it is foundthat uniform con- vergenceissufficient.Theproofofthisissimilartotheproofofcon- tinuity of an infinite seriesofcontinuous functions,Theorem 5-2. The example given earlier does not meet the conditions ofthe theorem. The integral roosin zx dx }ox convergesuniformly forz~Z'> 0,but not forz~o. 8-9.MTest forUniformConvergenceofan Improper Integral of the FirstKind.Sinceuniformconvergenceofanimproperintegralocca- sionallyarisesas aneededcondition,it isimportant to beableto test a givenintegral forthisproperty.The situation ismuchthe same as for series, and so for an intuitive understanding you are referred to Theorem 5-4. The caseofan improper integral ofthe firstkind isdealt withby the followingtheorem: Theorem8-11.Let M(x)beafunctionwhichispositive forallxand such that If(x,z) 1~M(x) forall z in aregionR.Then,if fa"M(x) ck exists,it can be concluded that fa"f(x,z)dx converges uniformly with respecttothe infinite limit, forz in R. 252COMPLEXVARIABLESANDTHELAPLACETRANSFORM 8-10.A Theorem forTrigonometric Integrals.The integrals IG" f(x)sinyx dz IG" f(x)cosyx dx are functions of y.In the proof of the Fourier integral theorem weshall needtoknowthelimitapproachedbythesefunctionsasybecomes infinite.Anintuitivepreviewoftheanswercanbegleanedfromthe observation that,as yincreases,the areas ofsinyx and cosyx,over any finiteintervalofx,approachzerobecausetheperiodapproacheszero. Ultimatelythisactionbecomesdominant,causingtheintegralsto approach zero. Inprovingthattheseintegralsapproachzerowearefacedwitha slightlycomplicatedsituationbecausetheoscillatorynatureofthe integrand is essential to the behavior of the integral, and soan appraisal mustbeusedwhichdoesnotdependupontheabsolutevalueofthe integrand.Weshalloutlinetheproof,assumingthat f(x)ispiecewise continuousbetween aand b.It ispossibleto subdividethe intervalof integration into Nincrements with end points Xkand to establish aset of constants AI:whichwill determinethe staircase function l Al As g(x)=... ANXN_l< x< b whichwillapproximatef(x)inthesensethat,foranysmallarbitrary positivenumber E,it willbetrue that ["E }aIf(x)- g(x)1dx< 2 For allvaluesofyit istrue that IJ.." f(x)sinyx dx I-I J.." g(x)sinyx dx I ~11" [f(x)- g(x)]sinyx dz I~1" If(x)- g(x)1dz< ~ and fromthi3wehave IJ.." f(x)sinyx dx I< ~+ IJ.." g(x)sinyx dx I THEOREMSONREALINTEGRALS But the integral onthe right issubject to the appraisal N Ii b g(x)sinyx dx I = IL: At /"","'+' sin yx dx I "-1 N 253 = IL: At cos YXt-y cosYXk+lI < 2 ~ f "-1 whereMisan upperboundofthe set ofAt numbers.The numbers N and Mare fixed,and so if Iyl> 4NM E wehave 2NM< ~ Iyl2 and thereforeIftJb I(x)sinyx dx I < E Since Eis arbitrarily small,this showsthat the integral approacheszero. Obviouslythe same result wouldhavebeenobtained if cosyx had been used.This completes the proof for the case where I(x)is piecewise con- tinuous and the limits on the integral are finite.However,the proof can readily be extended to include improper integrals if I(x) is APC and if it is absolutelyintegrabletoinfinity.SupposethattheAPCfunction I(x) becomes infiniteat Xoand that fo"II(x)1dx exists.Then,by virtueofthedefinitionofconvergenceofanimproper integral,wecan choosenumbers XI,X 2,and X a,where and such that 1/:'!(x) SinYXdXI< 1:"I!(X)ldX Roand-11"/2~8~11"/2and IH(Re i8 ) I Roand-11"/2~8~11"/2.Beyondthispointtheproofsfor the twocasesaredifferent. Consider case a;making the variable change z=Re i8 dz=jRei 8 d8 weget(H(z)dz=-j 10:/2Re i8 H(Re i8 ) d8 1 e.-0:/2 and so,invoking relation(8-26), I ( H(z)dz I ~10:/2RIH(Rd 8 )1d8< 1I"E(8-28) le.-0:/2 ifR> R o ,whereRoisthevaluedefinedinrelation(8-26).Since 1I"Eis arbitrarily small,it is established that lim(H(z)dz=0(8-29) R--:,.oolet Proceeding in asimilarway forcaseb,wehave (H(z)eb'dz=_jI0:/2Re;8H(Re;8)ebRco.8eibRalD8d8 le,-0:/2 andI { H(z)eb'dz I ~10:/2RIH(Re;8) le bR co. 8d8 le,-0:/2 (8-30) Nowlet Rbegreater thanR o ,sothat!H(Re;') I < E,byrelation(8-27). THEOREMSONREALINTEGRALS 257 Also,note tha.t the exponential is an even function of 8,so that for R> Ro wecan write IIe. H(z)e b •dz I ;;;;;ZRE10.. / 2 e bB - B dO At this point werefer to Fig.8;.9foran estimate ofcos0,namely, 2 cos8;;::1--0;;::0 - 'If'- o9- i" (8-31) FIG.8-9.Replacingcos9 by alinear functionwhich isalwayslessthan cos9. Therefore,assuming b< 0, 0 ~ 8 ~ ; This estimate isused onthe right sideofrelation(8-31)to give 10 ,,/2100:12'If'Ee bR 2REeb Bao .' dO~2REe bR e- 2bB8 /0:dO= -- (e- bB - 1) o0-b _'If'E(1bR)'If'E -TbT-e 1 and relate your findingsto the convergence properties obtainedin Prob.8-9. 8-11.Referring to Prob.8-9,compare the following: (a)fo Idyfo'"1Ie- o 'dx (b)fl2 dyfo'"ye- o •dx and and and relate your findingsto the convergenceproperties obtained in Prob.8-9. 8-12.For each of the followingcasescheckwhether or not thequestionmark can be replaced by an equality sign: Ca)foldyfo'"e-' sin xy dx?fo'"dx fol e-· sinxy dy (b)foldyfo"xe- O 'd;J;?fo'"dx folxe- O •dy Relate your conclusions to convergencepropertie!t of the integrals. 8-13.If fez)has the property that fo"lI(x)1dx 266COMPLEXVARIABLESANDTHELAPLACETRANSFORM converges,prove that fo" I(x+ y)dz converges unifonnly,a~y~Yo,wherea and yoare arbitrary real numbers. 8-14.For eachof the followingintegrals,usetheMtest to detennine arange ofy forwhich the integral fo" I(x,y)dz converges unifonnly: (a)I(x,y)=einxXyr e"" (b)f(x,y)=cos(x"+yZ)e-(s+tl) (c)I(x,y)=X Z + y" 8-15.Let I(x)be APC,and assume that fo"I/(x)l" dz converges.Useappropriate theorems in the text to prove that lim(.. I(x+ y)/(x) dz=(00I/(x)IO dz y-+oJoJo 8-16.Usecontour integration to evaluate the followingintegrals: f oosiny (a)_001 +y"dy f 00e;' (b)_..1+ y" dy J OOdy (c)_ 00(1+ yO)" , 8-17.Prove the identity (OOsiny+ycosY d =(..~ d z Jol+y"YJol+x 8-18.Evaluate theintegral (00dx JoVZ(1+ XZ) by the followingtwo methods. (a)Contourintegrationontheintegralasitstands.(HINT:Putabranchcut along the positive real axis.After obtaining the answer in this way,decidewhether abranch cut along the negative real axis could have been used.) (b)Letz=8 2 ,andobtainanewintegral,whichisthenevaluatedbycontour integration. 8-19.Evaluate the integral (0 00 .....,=--dz__ J(~ x(1+ x) where~is positive.(HINT:Use a branch cut along the positive real axis, and then decide whether or not you could get the same resultby using abranch cut alongthe negative real axis.) 8-20.Use the principle of contour integration in the complex plane to evaluate the integral roosin ... yd Joy(l- yO)Y THEOREMSONREALINTEGRALS 8-111.Verify both ofEqs.(7-54)forthe function 1 f(8)=1+ a 267 (HINT:UsetheprinciplesestablishedinSec.8-12,withasemicircularindentation around the poleat y= "'.The PV. integraldoesnot includetheintegral over this semicircle,and sothe portion due to it must be subtracted out.) 8-22.Verifyboth of Eqs.(7-54)forthe function 1 f(8)=(1+ s)2 (Seehint inProb.8-21.) 8-23.Let Z(s)be the impedance of acapacitor in parallel with aseries RL branch. Take eachelementvalueasunity.Therealcomponent of this impedance(forreal frequencies)is u(",) 1 1- ",'+ ",< and the imaginary componentis v(",) ",' - ",' + ",< Show that both ofEqs.(7-55)are satisfiedinthis case. 8-24.DefineasemicircleC'centeredat the originandofradiusR,and givecon- ditions on H(z)and specifytheorientationof thesemicirclesuch that lim(H(z)e" dz=0 R-+oole' wherec is the complexnumberIcle h . 8-26.Asan alternative to the method givenin the text forfinding ("'sinyd }oyy show that (..siny d- lim1, (J -, e;·dy+ f" e;vd y ) }o-y- y- ...... 02)_..Y•Y and then useJordan's lemma and the resultsofProb.5-33toevaluatethequantity on the right. 8.,26.LetarationalfunctionF(s)haveatleastasecond-orderzeroatinfinites. This means that the degreeof thedenominatormustbeat least2greater than the degreeof the numerator.Use an appropriate theoremto prove that the summation ofthe residues over all the poles iszero. 8-27.In the discussion leading to Theorem 8-7,the followingproperties of iterated integrals over the rectangles in Fig.8-7 are stated: {IIIH.y dy(:..+lUI~ x ()1'"< (constant)(.1y)2-" )1/1J "'IX- XI- IY - YI ( ~ + ~( ~ + l U~ JII'd y }",.Ix- x.- k.(y- y.)lnlx- x.- k.(y- y.)lp < (constant) (.1y)2-n- p Carry out theseintegrations,establishing the correctness of the above results. CHAPTER9 THEFOURmRINTEGRAL 9-1.Introduction.Inthischapterwedealwithperhapsoneofthe most important single topics in the theory ofthe Laplace transform.In Chap.1 you were given a nonrigorous development which formally states thatundercertainconditions,givenafunctionJ(t),wecanformthe function ff{jw)=f-''''",!(t)e-;..t dt and then recover J(t)by the inversion formula J(t)=2: PV J-"'..ff{jw)e;"tdw (9-1) (9-2) This is a statement ofthe bare essentials ofthe Fourier integral theorem. In theseformulaswand tarereal,andsothesearerealintegrals.In linear system analysis,t isusuallytime,but ofcoursethis isincidental, and t is regarded as merely arealvariable in this discussion. Withthebackgroundofthepresentchapterwearenowpreparedto give aproof which is mathematically more rigorousthan the proof given in Chap.1,and also weshall have aprecise statement of sufficient condi- tionson J(t)to ensure validity of the Fourier integral theorem. 9-2.Derivation of the Fourier Integral Theorem.Sufficient conditions willbedevelopedasweprogress;andthentheresultswillbecollected and stated as atheorem at the end.The firstcondition isthat J(t)shall be APC and the integral J_....IJ(t) I dt shallexist.Then,by Theorem 8-11,the integral J_..",J(t)e-;"t dt willconvergeuniformlywithrespecttotheupperandlowerlimits,for all w. The proofofthe theorem consistsin proving Eq.(9-2)to be true.As afirst step we substituteff(jw),as given by Eq.(9-1),into the integral of 268 THEFOURIERINTEGRAL269 Eq. (9-2).By omitting the factor 1/27r, the right side of Eq. (9-2) becomes PV J"ff(jw)eiw l dw=limJAeiw ' dwf 00dr -10.4-+00-A-110 The pair ofiteratedintegrationscanbeinverted,by virtueofTheorem 8-8,giving PV fooff(jw)eiw ' dw=limfoofer)drfAei.. dw -coA--+oo-00-A =lim2fOOfer)sinA(t - r)dr A_oo-00t- r =lim2foofeu+ t)sin Au du(9-3) .4-+00-00U The last step isarrivedat by changingthevariableinaccordancewith u=r- t and regarding t as constant. Now assume that t is a value at whichf(t-) andf(t+) exist.We write this integral in threeparts, as follows: f oesin Au 2_00feu+ t) -u- du=I1(A,t)+ 12(A,t)+ 13(A,t) f - a sinAu whereI1(A,t)=2_00feu+ t)-u- du 1 2 (A,t)=2+ t)sinuAu du 1 3 (A,t)=2hOOfeu+ t)sinuAu du (9-4) (9-5) The Riemann theorem for trigonometric integrals (Theorem 8-12)applies toI1(A,t)and1 3 (A,t)becausefeu+ t)/uis absolutelyintegrableover intervals whichavoid theorigin.Now wehave limI1(A,t)=0 A_oo limI a(A,t)=0 A_oo and so,fromEqs.(9-3)and (9-4),weare left with PV fOOff(jw)eiw ' dw=lim2f' feu+ t)sin Au du(9-6) -00A_oo-.U If f(t)is discontinuous at the value oft in question,then f( u+ t)isdis- continuous at u=0 and this would lead to difficulty in alater step if the integral were left in this form.This troublecan beavoidedby writing it as the sum of two integrals, 270COMPLEXVARIABLESANDTHELAPLACETRANSFORM 2 feu+ t)sin Au du= 2fOf(u+ t)sinAu du u u + 2 feu+ t)sinAu du }ou sinAu =2[feu+ t)+ f( - u+ t)]--du ou (9-7) Now changethe variableto w=Au, giving 2+ t)sinuAu du= 2 !oM [f(t + X)+ f(t - X)] si: Wdw Let Aapproach infinity,without justifying the step,to get aclueto the answer.Justificationwillcomelater.Thisyieldsthetentativeguess that the right-hand sideofthe aboveequation is 2[f(t+)+ f(t-)]r" sin Wdw=7r[f(t+)+ f(t-)](9-8) }ow Now weshallprove that this iscorrect. The sine integral isthe starting point,and weproceed by modifying it as follows: 10 ...10..1.&.10&.A SInWdl'sm Wdl'smud --W=Im--W=Im--u oW..1.--..0W..1.--..0U Therefore,thetermontheleftofEq.(9-8)canalsobewrittenas follows: lim2 [f(t+)+ f(t-)] sinAu du ..1.--..}oU A check on the correctness of the tentative answer given by Eq.(9-8)is obtained by computing the differencebetweentheaboveexpressionand therightsideofEq.(9-7).Byomittingthefactor2,thisdifference IS lim[r& fCt+ u)- f(t+)sin Au du+ f(t- u)- f(t-) sin Au dU] A--"}OU}oU Asuapproacheszero,eachofthequantitiesf(t+ u)- f(t+)and f(t- u)- f(t-)approacheszero,becausef(t)ispiecewisecontinuous. Thus, there isthe possibility ofthe above quantities,when divided by u, beingintegrableovertheintervalindicated.However,theywillbe integrable only if the numerators approach zero as fast as, or fasterthan, u. THEFOURIERINTEGRAL271 Asasufficient condition assume that wecan findnumbers Kand asuch that when IJ(t+ u)- J(t+)1< K1u"" IJ(t- u)- J(t-)I< K2u'" u< 8 (9-9) and where al and a2are each greater than zero.This willensure integra- bility.For example,then, I e J(t+ u)- J(t+) du I ~(8 K 1 u",,-1du=K 18'" JouJoal andsimilarlyfortheother integral.Relations(9-9)arecalledLipshitz conditionsoforder aand are aproperty of J(t)whichwenowrequire,in additionto theoriginalcondition ofbeingAPe. Assuming that conditions (9-9)are satisfied,Theorem 8-12can now be used to arrive at the results lim A .... "lo 8 J(t+ u)- J(t+).AdO '-..:.------'----"---"--'---'--'- SInuu= ou eJ(t - u)- J(t-) sinAudu=0 Jou lim A .... " This confirms the earlier guess that the right side ofEq.(9-8)is acor- rect evaluation of the right side of Eq.(9-6),and therefore alsoofthe left sideofEq.(9-3).Inotherwords,wehaveproved,fortheconditions assumed,that J(t+)+ J(t-)= -.!. PVf"fJ(jw)e;"t dw 2211"_" (9-10) at anypointwhere J(t)satisfiesLipshitzconditions ofordera> O.Of course,at anypointwhere J(t)iscontinuous,(f(t+)+ J(t- )]/2= J(t), and soEq.(9-2)isan adequate representationif J(t)=J(t+)+ J(t-) 2 isused forthe definitionof J{t)at points offinitediscontinuity. A fewwords about the physical meaning of the Lipshitz condition may beenlightening.Existenceofright- andleft-handderivativesat each valueoftwouldbesufficienttocarryouttheconcludingstepsofthe proof,without explicitly stating the Lipshitz condition.In fact, existence ofthesingle-sidedderivativeisequivalenttotheLipshitzconditionof order 1.Most practical functions,like the examples in Fig.9-1,do have right- andleft-handderivatives.However,wewouldliketobeabit 272COMPLEXVARIABLESANDTHELAPLACETRANSFORM moregeneraland include functionslikeFig.9-2.Here there arepoints wherethe function remains finite,mayor may not bedisconti:ijuous,but wherethesingle-sidedderivativebecomesinfinite.Thesefunctions FIG.9-1.Examples of functionshaving finitesingle-sided derivatives. FIG.9-2.Examplesofboundedfunctionswhichhavederivatives(orsingle-sided derivatives)which become infinite. satisfy the Lipshitz condition, and so the Fourier integral theorem is valid forthem.'" Wenow state the Fourier integraltheorem as follows: Theorem9-1.Letf(t) be a function which is almost piecewise continu- ous,whichsatisfiestheLipshitzconditionofordera> 0at eachpoint wherethe functionisfinite,and forwhichthe integral f_"'",.lf(t)1dt converges.Then, ff(jw)=f_..",f(t)e- iwt dt defines the function ff(jw)by an integral which converges uniformly with •The function1/ v' -log It I doesnot satisfy aLipshitz condition at the origin. THEFOURIERINTEGRAL 273 respect to the infinite limits, for all real w.Furthermore, f(t)is related to by f(t+)+ f(t-)=PV fOGdw 22"11"_ .. atallpointswheref(t)isnotinfinite.Thefirstintegraliscalledthe Fourier integral, and the second integral is called the inversion integral. 9-3.Some Properties of the Fourier Transform.For further appraisal ofthe Fourier integral theorem weconsidertwo examples.Asthe first one,take the evenfunction forwhich f(t)=e- alll a> 0 =fOe(a-i.. )1dt+ { '"e-(a+iw)ldt -00Jo 11 = a- jw+ a+jw 2a a 2 + w 2 Asthe second example,consider theoddfunction which gives { _eal f(t)=e- al t< 0 t> 0 =- J0e(a-iw)ldt+ ( 00e-(a+iw)ldt -00}o 11 ---+-- a- jwa+ jw .2w - J a2+ w2 (9-11) (9-12) (9-13) (9-14) In each casenote that is an analytic functiongiven by acompara- tively simple formula which has meaning forall values of w,realor com- plex.Ontheotherhand,ifw=x+ jy,theFourierintegralbecomes· J_'","f(t)e" l e-;:'Idt Introduction ofthe factore" 1 may prevent this integral fromconverging certain values ofy.Thus, although the integral may not converge for all complex values of w,weseem to get a function from the integral which is defined for all values of w,as well as real.At least this is true forthe examplesgiven. The above factspoint up the reason formaking adistinction in termi- nologybetween the Fourierintegralandthefunctionff(jw).The latter * Hereweare making an exception to the practiceotherwisefollowedin this text, that Colshall bereal. 274COMPLEXVARIABLESANDTHELAPLACETRANSFORM iscalledthe Fouriertransformoff(t).It isafunctionof w in whichno integralneedappear.Thetransformisafunctionwhichcanberepre- sented by the integral for certain ranges of complex w but which can exist for other values of w. We shall now briefly consider a few of the universally important proper- ties of the Fourier transform function ff{jw),as they are related to proper- ties of f(t). 1.Properties of Real and Imaginary Parts offf{jw).In the first example f(t)==f( -t),andwefoundthatff{jw)isarealfunctionofwhaving thepropertyff{jw)==ff( -jw).Inthesecondexamplef(t)==-f( -t), andthecorrespondingff{jw)isimaginaryandhastheproperty ff(jw)==-ff( -jw).Thesearespecificexamplesofageneralproperty whichcanbederivedby writing f(t)asthe sumofevenandodd parts: f(t)=f.(t)+ f.(t) where f.(t)=f(t)+l( -t)f.(t)=f(t)-l( -t) Thenthe Fourierintegralcan be written J 00f(t)e- iwt dt=(oo[f( -t)e iwt + f(t)e- iwt ] dt -0010 ff(jw)= 2 fo 00f.(t) cos wt dt- j2 fo"f.(t) sin wt dt (9-15) (9-16) (9-17) Thus, whenf(t) is real, the real part of ff{jw)is always an even function of w,and the imagmary part is always an odd functionofw.Furthermore, iff(t)iseven,ff(jw)isreal(andalsoeven),andiff(t)isodd,ff(jw)is imaginary (and also odd).Agoodway to summarize these properties is to write ff(jw)=ff( -jw)wreal(9-18) In the terminologyofChap.7,ff{jw)isarealfunctionof jw. This general case, for a real functionf(t),can be summarized by defining twotransform functions,as follows: ff{jw)=ffr(jw)+ jff,(jw) (9-19) whereffr{jw)=2fo"f.(t)coswtdt ff,(jw)=- 2fo"fo(t)sinwtdt (9-20) 2.Differentiability of theFourierTransform.If f(t)satisfies the condi- tions of the Fourier integral theorem, as stated by Theorem 9-1, weknow that THEFOURIERINTEGRAL 275 converges uniformly for all w.Uniform convergence is sufficient to allow us to write ff{jw)- ff(jwo)=J_.... f(t)(e-Jo>.- e-Jo>o')dt and to take the limit as wapproaches wooThis isbyvirtue ofTheorem 8-10.The integrandapproacheszero,and soit followsthatthetrans- formisacontinuous functionofthe realvariable w. Now apply the Fourier integral theorem to the function tf(t), andadopt the notation ffl{jW)=J_....tf(t)e-Jo>'dt This functioncanbeintegrated under the integral with respectto w,by Theorem8-8,and sowecan write ('" fftUU)dU= jJ"f(t)(e-Jo>'- 1) de )0- .. = j[ff{jw)- ff(O)] Nowdifferentiatewith respect to w,to obtain ff(.). dff(jw) 1JW= J ----a;;;- (9-21) Since ffl{jW)exists and is continuous,weseethat ff(jw)has acontinuous derivativewithrespecttotherealvariablew.Asimilarresultcanbe obtained if t 2 f(t) satisfies the conditions of the Fourier integral theorem, in which case the second derivative of ff{jw)is found to be continuous.We summarize by stating that if tnf(t) is absolutely integrable from- 00to00, thenthenthderivativeofff(jw)withrespecttow iscontinuousandis givenby d"ff{jw)=~ff,,(J'w) dw"J" (9-22) Intheabove,wmustberealbecauseuniformconvergenceisassured only forreal w.Thus, in Eq.(9-21)wecan say that the derivative exists for real w but not necessarily for complex w.Consequently, this equation doesnotestablish that ff(jw)isanalytic. Toshowthat ff(jw)isnotnecessarilyanalytic,considerthefunction sin bt f(t)=t(t2_'/r 2jb2) By routine integration,the Fourier transform is foundto be o ~Iwl~b Iwl> b 276COMPLEXVARIABLESANDTHELAPLACETRANSFORM For this example,ff(jw)has acontinuous firstderivative,but the second derivativeisnotcontinuous,andhigher-orderderivativesdonotexist. Ifwechecktheabsoluteintegrabilityoftnf(t),wefindthatabsolute integrabilityisretainedwhenn=1butnotwhenn=2.Hence,it is to be expected that the firstderivative will be continuous but not neces- sarily the second. Thecasewheref(t)isidenticallyzeroforIt I greaterthansomefixed value isofparticular interest.For such afunctiontheFourier integral has finitelimits,and so t"f(t)is absolutely integrable for all n.We con- cludethat in this case its Fourier transform is indefinitely differentiable with respectto w. Many other properties of the Fourier transform function can be derived, but the same information can beobtained more easily fromthe Laplace transform,because the theory of functionsofacomplexvariable is then moreextensively applicable. 9-4.Remarks about Uniqueness and Symmetry.For agiven f(t),its transformff(jw)isuniquelydeterminedbytheFourierintegraloff(t). Therefore,eachFourier integrable functionhas oneand onlyonetrans- formfunctionff(jw)asits"mate."Wenowaskwhetherornottwo differentf(t) functions could ever produce the same ff(jw) function.Sup- pose that/1(t) andh(t) are two different functions but that the transform ofeach isthe same function ff(jw).Sinceff(jw)is the same for each,an identical i n v ~ r s i o nintegral is obtained for /1(t)andf2(t).Therefore, from Eq.(9-10)it followsthat fl(t)and f2(t)must satisfy the equation fl(t+)+ /1(t-)_h(t+)+ h(t-)(9-23) 2- 2 At any continuouspointofafunction f(t), f(t+)+ f(t-)=f(t) 2 and thereforeEq.(9-23)tellsusthat /1(t)=: f2(t) (9-24) except possiblyat isolatedpoints whereoneortheother functionmight bedefinedarbitrarily.TheinversionintegI'alcangivenoinformation aboutwhetherornotfl(t)and h(t)areequalatsuchisolatedpoints. This is to be expected,becausethe setofsuch isolatedpoints is aset of measureO.WerecallfromTheorem8-5that an integralisunaffected if the integrand is defined arbitrarily over aset ofmeasurezero.Thus, theFourier integralcan yieldthe same ff(jw)functionfortwo f(t)func- tions which differ over aset ofmeasure 0,and the inversion integral will beinsensitivetothisdifference.Thispossibledifferencebetweentwo functionshavingthesametransformisprimarilyofacademicinterest. THEFOURIERINTEGRAL 277 In fact,if wecontinue to agreeto defineafunction at adiscontinuity as the mean of the limits approached from the two sides, thenf(t) is uniquely related to ff(jw). This property ofuniquenessisvery important.It means that tables ofpaired functionscan be constructed which are solutions ofthe pair of integral equations ff{jw)=f-....f(t)e-;"Idt(9-25a) f(t)=2 ~PVf-....ff(jW)e"'"dw (9-25b) whereEq.(9-24)isunderstoodtodefinef(t)inthesecondequation. Becausetheserelationshipsoccurinpairs,andbecausewehaveestab- lisheduniqueness,it followsthat if oneoftheseintegralsisknownthe other one is automatically known.For example, in Sec. 8-9 we obtained the formula --- =e-aIl1e-;"1 dt2af" a 2 + w 2 - .. (9-26) In view of the above property we immediately know the value of an addi- tional integral,namely, e-GIII=- --- ei",tdw 1f"2a 2'IT_ ..a 2 + w 2 (9-27) TheusualPVdesignationisnotneededherebecausetheintegralcon- verges as it stands. In thiswayweseethattheFourier integral theorem has the effectof doubling the size of a given table of integrals like Eq. (9-25a).If we have atableofintegralsrepresentingintegralslikeEq.(9-25a),wealso implicitlyhaveatableofintegralslikeEq.(9-25b).Exceptwherethe PVdesignationisneeded,thesetwointegralsareessentiallythesame. Letusconsiderthisstatementabitfurther.Equation(9-27)canbe made to look like Eq.(9-25a)by replacing w by t and t by-w to give !: e- al .. 1 =f"_1_ e-;"I dt(9-28) a_ ..a 2 + t2 If Eqs. (9-27)and (9-28)are typical of all cases, it would appear that to obtain onepair offunctionssatisfying Eq.(9-25a)istoobtain asecond pair.In this casethe two pairs are: 1(1)1f(i"') 6- 111 2a at + ",I 1 ~6-el..1 a ' + I' a 278COMPLEXVARIABLESANDTHELAPLACETRANSFORM Onenaturallyaskswhetherthis illustratesagenerallyvalidconclusion. Aswasmentioned above,the integralofEq.(9-25b)cannot bemadeto lookliketheintegralofEq.(9-25b)unlessthe PV designationcanbe omitted from the latter.Also,this example is aspecial case in the sense that fJ(jw)is real.Consider the general case of Eq.(9-25b),in which the integralconvergeswithouttakingtheprincipalvalue.Then,iftis replacedby-t, wecan write f( -t) =~f'"fJ(jw)e- j .,/ dw 2'11"-'" and ift and warenowinterchangedweget 2'11"f( -w)=f-"'",fJ(jt)e- j .,/ dt(9-29) If fJ(jt)is real,this last equation looks like Eq.(9-25a), except for inciden- taldifferences in notation.If fJ(jt)is complex,j( -w) is neither even nor oddand,inthenotationofEqs.(9-19)and(9-20),theaboveequation reduces to the pair 'II"[f( -w) + few)]=f-"'",fJ.(jt)e- M dt 'II"[f( -w)- few)]= jf-"'",fJ;(jt)e-jwt dt Each ofthese is similar toEq.(9-25a). (9-30a) (9-30b) No condition has been stated whereby we can know when an arbitrarily given fJ(jw)function willbethe Fourier transform of some f(t).In view ofthe symmetry properties mentioned above,wecan at least say that if f-"'",IfJ(jw)1dw converges, then fJ(jw)is the Fourier transform of somef(t) function.This givesnoinformationaboutthecasewheretheinversionintegralcon- vergesonly in theprincipal-value sense.In that case,if ~PVf'"fJ(jw)eiw/dw 2'11"_'" converges to a function f(t) , this function can then be tested to determine whether ornot it has aFourier transform. Theseremarksareofmorethanacademicinterest,aswecanseeby consideration of the following example.In system analysis we often deal witharectangularpulsedefinedby f(t)={ ~ itl< 1 Iti>1 THEFOURIERINTEGRAL Its Fourier transform isreadily foundtobe ff(jw)=2 sin w w 279 Thisfunctionisnotabsolutelyintegrable.Thuswehaveanexample where a function which is not absolutely integrable is a Fourier transform, emphasizingthefactthatabsoluteintegrabilityissufficientbutnot necessary.By invoking the ideas of symmetry, we can also conclude that sint t hasaFouriertransform.This functiondoesnot satisfy the sufficiency conditionsofTheorem 9-1. 9-6.Parseval's Theorem.Consider two functions f(t)and get)having the property that the integrals J_....lf(t)1dtand f _....Ig(t) I dt exist,and let one ofthese functions be PC and bounded forall t,and the other APC.From thesewedefineathird function ret)=f-....f(r)g(r+ t)dr (9-31) Ultimately weshall want the Fourier integral of ret),but first we investi- gateconvergenceofthedefiningintegral.Supposethatget)isthe bounded PC function.Since it is bounded, for all t and rwe can say that there isaconstant Msuch that and therefore Ig(r+ t)1< M If(r)g(r+ t)1< Mlf(r) I for all t.The term on the right is absolutely integrable, and therefore, by Theorem8-11,it followsthattheintegralinEq.(9-31)convergesuni- formly for all t.If f(t)is the function designated as being PC, we replace r+ t by u in Eq.(9-31)to give ret)=f-....feu- t)g(u)du whichcan be treated by asiInilar argument. Since Eq.(9-31)converges uniformly,wecan obtain r( co)by allowing ttobecomeinfiniteunder the integral.If g( co)exists it mustbezero, otherwiseget)wouldnotbeabsolutelyintegrable.Inthatevent r( co)=O.It can also beshown that ret)is continuous (seeProb.9-23). 280COMPLEXVARIABLESANDTHELAPLACETRANSFORM Weshallshowthat ret)hasaFouriertransformby investigating the integral J _....r(t)e-;"I dt=l ~J _ABe-;"I dtJ _....J(T)g(T+ t)dT(9-32) B-+ .. Byvirtueofuniformconvergenceofthelastintegralontheright,the order ofintegration can be changed,giving J-....r(t)e-;"I dt=1 ~ ~J-....J(T)dTJ_ABgeT+ t)e- iwI dt B-+ .. Now changethe variable ofintegration in the integral with respect to t, by making the substitution u=T+ t,as follows: J ..r(t)e-;"I dt=limJ"J ( T ) ~ TdTJ.A+Tg(u)e-;"u du(9-33) -110A ......ao-ao-B+T B-+ .. The integral of anonnegative functionover finite limits is not more than its integralover infinite limits.Therefore,forallvalues ofA, B,and T, it istrue that f A+Tf" -B+TIg(u)1du~_ ..Ig(u) I du and with this information we can write the following sequence of inequali- ties for the absolute value of the integrand of the Tintegral in Eq.(9-33): Theintegralontherightisconstant,and J(T)isabsolutelyintegrable. Therefore,by Theorem 8-11wecan say that the integral on the right of Eq.(9-33)converges uniformly with respect to either Aor B.According to Theorem 8-10, it is possible to place the limits shown in Eq. (9-33) inside the Tintegral,giving f ..T(t)e- iwt dt=J"J(T)eiw'dTlimf A+Tg(u)e-;"u du -aD-00A ......00-B+,.. B-+ .. =J _....J(T)eiw'dTJ _....g(u)e-;"u du If welet !I(j,.,)andg(j,.,)representthe respectiveFourier transforms of J(t)and get),we see that the two integrals on the right above are, respec- tively,!I( -j,.,)andg(j,.,).Wehavenowprovedtheinterestingresult (9-34) THEFOURIERINTEGRAL281 Theinversionintegralcanbeappliedtotheright-handsideofEq. (9-34)to give f_"'J(r)g(r+ t) dr=2 ~PV f-: fJ( -jw)g(jw)eiw l dw(9-35) * Since ret)is continuous Eq.(9-35)is valid for all values of t.For the par- ticularvaluet=0, f '"1f'" _J(r)g(r) dr=27rPV_'" fJ(-jw)g(jw) dw =2 ~I'V f-"'",fJ(jw)g( -jw) dw(9-36) This resulthasaparticUlarly significantphysicalinterpretation if f(t) andgCt)areidentical.ThisfunctionisnecessarilyPCandbounded, becauseinthederivationofEq.(9-36)weestablishedthat onlyoneof thetwofunctionscouldbeAPC.Wealsonotethat fJ( -jw)fJ(jw)isan evenfunctionofw,andthereforeanintegralofthisfunctioncannot depend upon odd-function properties tomake the principal value exist in theabsenceofordinaryconvergence.Accordingly,theprincipalvalue called for in Eq.(9-36)is not required.Equation (9-36)now becomes f '"1f" _'"[f(r)]2dr=27r_'"fJ(jw)fJ(-jw)dw (9-37) This can bewritten in another way by recalling that fJ( -jw) is the conju- gate offJ(jw),sothat fJ(jw)fJ( -jw)= IfJeiw)i2 andfinally f '"1f'" _'"[f(r)j2 dr=27r-00IfJ(jw)i2 dw (9-38) The result just derived issummarized in the followingtheorem: Theorem9-2.Parseval'sTheorem.If f(t)ispiecewisecontinuous, absolutelyintegrable,andbounded,andifitsFouriertransformis des- ignatedby fJ(jw),then fCt)and fJ(jw)arerelated by the formula f '"1f'" _00[fCt)]2dt=27r_'"\fJ(jw) 12dw •Equations(9-34)and(9-35)werederived on the assumption that one of the two functionsisPC,while the other may beAPe.It can be shown that these equations are still valid if both of these functions.are APC, if one of them remains bounded as It I becomes infinite.However, in that case, r(t)will be APC.In the present discussion wewant r(t)to be continuous,and soweadhere to the original conditions on f(t)and l1(t).This footnote is intended primarily for later reference, which occurs in Chap. 11. 282COMPLEXVARIABLESANDTHELAPLACETRANSFORM A physical interpretation of this theorem is possible.If f(t)represents atime-varyingphysicalquantity,under certain conditionsthe left-hand integral is a measure of the energy transfer.The right-hand integral is a summation ofthe energiesofthe differentialcomponentsoftheFourier spectrum of the function.Engineers are familiar with the Fourier-series counterpartofthis,thatthepowerassociatedwithaperiodicfunction equals the sum of the powers associated with its harmonic components. PROBLEMS 9-1.Determine whether or not the followingfunctionsmeet the conditions of the Fourier integral theorem: (o)/(t)- {k t l (c)I(t)=sin t t (e)I(t)=s i ~ t l, It I < 1 1< It I 9-2.Check the function I(t)={ ~ 1 (b)I(t)- {~ ~ t (d)I(t)=tRe- 1I (J)f(i)=t!O}; It I > 1 -1 1 -1 ~t~1 wheren iseven. (a)Showthatthisisanevenfunctionandthatthefirstn- 1derivativesare continuous. (b)Obtain the functionfJ(j",)fromthe Fourier integral. (e)Check whether or not(",)"fJ(j",)goesto zeroas ",becomes infinite,as would be indicatedby the result stated in Probs.9-12and 9-13. 284COMPLEXVARIABLESANDTHELAPLACETRANSFORM 9-111.UsingtheideapresentedinProb.9-13,andusingthefunctionsgivenin Probs.9-10 and 9-11, (a)Obtain the Fourier transforms of !Ct)- (1- t)e- itl andI(t)- (1- ItDe-I,1 (b)Usingthej",functionsobtainedinparta,recoverthegivenI(t)functions, through the inversion integral. 9-16.If I'(t)becomesinfiniteat tto,butinsuchawaythat I(t)isAPe,show that I(t)satisfies aLipshitz condition at to. 9-17.Showthat,if I(t)hasaderivativeatto,thenatthispointitsatisfiesthe Lipshitzcondition of order unity. 9-18.Show that the function I(t)=- 10: It! does not satisfy aLipshitz condition at t=O. 9-19.ReferringtothestatementoftheFourierintegraltheorem,wenotethat (sin t)/t does not meet the condition stated in the theorem.Nevertheless, it is found that 'this functionsatisfiesthe integralrelationshipstatedinthetheorem.Discuss whether or not the theorem isstated in such away as to permit this specialcase. 9-20.If I(t)ismade up of the finitesum .. I(t)=L n-l where each b n is real and positive, show that ff(j.,,)is a rational function having simple poles at j."='b.,b"etc. 9-21.Use the Parseval relation to evaluate the followingintegrals: (a)f-.... x)' dx J ..sin't (b)_..t'(t"_".IP dt 9-22.Use the principles established in Sec.9-5to evaluate the integrals J ..sin x (a)_ ..x(1+ x")dx f ..e-J"" (b)_ ..1 + x. dx 9-23.Prove that the function ret)=f _.... /('T)g('T+ t)d-r iscontinuous at each value oftunder the conditiontl given in the text. 9-24.Obtain theFourier transform for J(t)=Binbt t CHAPTER10 THELAPLACETRANSFORM 10-1.Introduction.TheFourierintegraltheoremoccupiesacentral position in the theory of linear integrodifferential equations.This fact is broughtout inChap.1,andthedevelopmentgiventheremaybecon- sideredmotivation forthepresent chapter. Threebasictasksliebeforeus.AsisstatedinChap.9,the Fourier integral theorem is restricted to functions which approach zero at t=± QO fast enough to make the Fourier integral converge.One task isto show that this restrictioncanberemoved.The secondtask isto extendthe Fourier integral theorem to those cases where wewant to investigate the responseofalinearsystemtoan excitationwhichcommencesat t=O. Thethird task isto developcertain propertiesoftheresultingmodified transforms,whicharerelabeledLaplacetransforms.Thepropertiesto be investigated are those which make the Laplace transform auseful tool in the solution oflinear equations. Another introductory note is iI'der.In Chap. 8 several theorems on realintegrationareenunciated,uuchasthetheoremonintegrationby parts.In Chap.10 weshall use these theorems,but frequently the inte- grand willbeacomplex functionofarealvariable.A complexfunction g(x)of arealvariable xcan always bewritten wheregl(X)andg2(X)arereal.Sincethesevarioustheoremsonreal integration apply to each ofthe functionsgl(X)and g2(X),the respective theorems apply alsoto g(x). 10-2.The Two-sided Laplace Transform.In Chap.9 wesaw that for certain functions f(t)the F o u r i ~ rintegral (10-1) exists and yields a transform which is acomplex function ofthe real vari- ablew.Nowweallowwtobecomplex,but forlaterconvenienceit is bettertoletjwbethegeneralcomplexnumber8.Then,inSopurely 285 286COMPLEXVARIABLESANDTHELAPLACETRANSFORM formalsense wecan replace jwby sin Eq.(10-1),giving 5'(s)= f _-..f(t)e-"dt (10-2) where the function 5'(8)is represented by the integral for any value of 8 for whichthe integralconverges.5'(8)isthe two-sidedLaplacetransform of f(t),and the defining integral is the two-sided Laplace integral.A detailed considerationofconvergenceofthisintegralwilltakesometimeto develop.Initialinsightisgainedbywriting8=u+ jw,SOthatEq. (10-2)becomes 5'(s)=1-....f{t)e-vle-;"Idt which is theFourier integralofthe function f{t)e- vi (10-3) FromtheFourierintegraltheorywecansaythatatleastasufficient condition for the existence of the integral in Eq. (10-3)is that the integral f-....If(t) Ie-vIdt shall exist. Weimmediatelysensethatforagiven f(t)thisintegralcanexistfor certain values of ITand not for others.In particular, it may converge for certainf{t)functionswhicharenotthemselvesabsolutelyintegrable. This wouldbethe case forthe function f{t)={ !,; ~g forwhichf _....f(t)e- vi dt=f ~..e(l-v)1 dt+ fo"e- vi dt The firstintegral on the right converges if IT O.Therefore,the combination converges if0< IT< 1, showing that the integral of Eq.(10-2)converges in a vertical strip in the 8plane.Later on it willbe shown that this is the general situation, that in allcases the integral of Eq.(10-2)converges in avertical strip.How- ever,this strip may range fromthe wholeplane down to asingle vertical line,depending on the nature of f(t).The function sin bt f(t)=t(t2- r2jb2) which appears as an example in Chap.9,is acase wherethe strip ofcon- vergenceisreducedtotheimaginaryaxis.Later onweshallfindthat, if the integral in Eq.(10-2)converges in astrip of finitewidth,then 5'(8) is an analytic functionof 8. THELAPLACETRANSFORM 287 Wecould goon to develop adetailed analysis of the two-sided Laplace transform.However, abetter procedure is to recognize that the defining integral can bewritten in twoparts,as follows: f _....f(t)e-" dt= f ~..f(t)e- a ,dt+ fo" f(t)e- a 'dt =fo"f( -t)e"' dt+ fo"f(t)r a ,dt Therefore,it willbesufficientto study the singleintegral fo"f(t)e- a 'dt Havingdonethis,withdueregardforsignchanges,wecanapplythe resultsto J 0f(t)e-"dt=( ..f( -t)e d dt -..Jo and in this way wecan get the information wewant about the two-sided Laplacetransform fromproperties ofintegrals from0 to00. 10-3.FunctionsofExponentialOrder.Inaseriesofstepsweshall investigate convergenceproperties ofthe integral F(s)= fo"f(t)e- a 'dt(10-4) F(s)is called the one-sided Laplace transform of f(t) , or merely the Laplace transform.It is represented by the integral in Eq.(10-4),which we shall call the Laplace integral, for all values of s for which the integral converges. As a first step a new class of functions is defined.Let f(t)be APe, and let it have the further property that there is a real number ao such that limf(t)e-a.'=0whena> ao (10-5) 1-+" and withthe limit not existingwhena 0 1-... In fact,t n isofexponentialordero.In an unstablesystemafunction may increase as (3"',and weseethat limea'e- a '=0 ,-+,. ifa> a.Thus,the functione a 'is ofexponentialorder a. The order number aowillbe- 00for all functions which are identically zerobeyond some finitevalue oft.Thus,wemay expect aoto lie in the range - o o ~ a o < O O(10-6) 10-4.TheLaplaceIntegral forFunctionsofExponentialOrder.We nowconsider convergence ofthe integralin Eq.(10-4)when f(t)is APe and EO,ao.From Theorem 8-6weknowthat fo"f(t)e- BI dt converges if fo"If(t)e-"Idt=fo"If(Ole-v'dtu=Res converges.Convergence of the integral on the right willbe investigated for uin the range ao< u(10-7) For any uin this rangewecan pick anumber a2such that ao ao. For uniform convergence, an integrable function independent of s (oru) isneeded for use inthe test given in Theorem 8-11.Let al be anumber greater than ao,and let ube inthe range (10-9) Foranychoiceofal wecanfindanumbera2suchthatao Re 80.The proof requires an auxiliary function wet)=f," dT(1(}'1O) This isacontinuous functionoft,and its derivative is w'(t)=-f(t)e-'" and in terms ofthisthe Laplace integral can be written * Obviously the APe condition is for ts:;;O.Weomit this qualification in this and followingtheoremsbecauseitwouldamounttoatrivialredundancy,integration being always from0 toco. 290COMPLEXVARIABLESANDTHELAPLACETRANSFORM wherethecomplexvariable8hasbeenreplacedby8=80+ z.The Cauchy principle ofconvergencewillbeusedto establishconditions for convergence of the integral on the right ofEq.(10-11).Thus, weare to show that corresponding to an arbitrary small E> 0 we can find a number Tosuch that I!(t)e-"dt I =Iw'(t)e-"dt I< E (10-12) whenA', A> To Theaboveintegralsatisfiestheconditionsforintegrationbyparts, given in Theorem 8-4,subject to the comments given in Sec. 10-1 relative to e- al being acomplex functionofthe realvariable t.Thus, w'(t)e-"dt= + z w(t)e-.t dt =-w(A')e- aA '+ w(A)e- aA + z w(t)e-"dt(10-13) The absolute value of the right-hand sideofEq.(10-13)is lessthan the sum ofthe absolutevalues ofindividualterms.Therefore, I!(t)e-"dt I To,wehave Iw(t)1< E' t> To Iw(A')I,Iw(A)1< E' andifx> 0andifAisthelargerofAandA',thenrelation(10-14) becomes I !(t)r d dt I< E'[ e- zA '+ e- zA + 1;1(e- zA '- e- zA ) ] which in turn is lessthan e' (2 + 1;1)=E (10-15) This canbethe Eofrelation(10-12). THELAPLACETRANSFORM 291 For any fixed value of z,with x>0,the above quantity in parentheses isfinite,andbymakinge'smallenoughthewholequantity(E)isarbi- trarily small.Since E'determinesTo,weseethat fo'"f(t)e- d dt converges when Re s> 0"0 where0"0= Re so.ThisconditiononRe scomesfromthecondition x>0,sincex=Re z=Re (s- so). The above discussion provides the range of s for ordinary convergence. Thisisnottherangeforuniformconvergence,becauseTois dependent upon Izl,through Eq.(10-15) and the dependence of Toon l.In order to get the region of uniform convergence, let 8 be the angle of z,and observe that 1;1=Ico! 8\ whenx> 0.If 8 isrestrictedto the range 181~8'Re So Furthermore, convergence ofthe Laplace integral is uniform with respect to the upper limit when 'If' lang(s- so)1~8'< 2 292COMPLEXVARIABLESANDTHELAPLACETRANSFORM NotethatwedonotgetconvergencewhenRe 8=uo.Thismeans that, although the integral converges when s=So,it does not necessarily converge for some other s having the same real part as So.A simple exam- ple can begiven to illustrate this point.Let f(t)~{{ Then for 80=0+ jwothe integral O;:;;t 11'".Regions ofuniform convergence are illustrated in Fig.10-2. Up to this point we have been considering convergence with respect to the infinite limit.In addition,since f(t)may be APC,it isnecessaryto consider convergencewith respect to points where f(t)becomes singular. TheAPCcategoryoffunctionsisdefinedinsuchawaythat,ifisa singular point,the integral f(t)dt exists,wherechand chare arbitrary small positive numbers.If 11'1is any realnumber,weseethat (bH'lf(t)leIClllt dt;:a;eICllIO.+',>(t'Hl1f(t)1 dt THELAPLACETRANSFORM alsoconverges.If Re8~Ul, If(t) e-otl=If(t) le-R. Ca)t~If(t) le l " 1lt and soby Theorem 8-11it followsthat 295 convergesuniformly forRe 8~U1.Since U1is any number,the Laplace integralconvergesuniformlywithrespecttofinitesingularpointsof f(t)for8in any righthalf plane. 10-7.ConvergenceoftheTwo-sidedLaplaceIntegral.InSec.10-2 thetwo-sidedLaplaceintegralwasshowntobethesumofthetwo integrals J 00f(t)c,t=r" f(t)c sC d' +r" f ~ - t ) e "dt -00JoJo €10-17) The two-sided Laplace integralconverges if each ofthe two integrals on therightconverges.Thefirstoftheseconvergesin arighthalfplane. Thesecondintegralissimilarbuthastheexponent+8 insteadof- 8. Thismakesthesecondintegralconvergeinarighthalfoftheminu8 8plane and thus in a left half of the 8plane.The given integral will con- vergeif these two half planes overlap. To pursuethis in moredetail,supposethat 10 ~f( -t)e- at dt hasan abscissaofconvergence-U.2.It converges for Re 8>-U.2 Thenfo" f( - t)e,t dt converges for-Re8 =Re(-8)>-Uc2 orRe 8< U.2 NowsupposethatthefirstintegralontherightofEq.(10-17)hasan abscissa of convergence Ue1.Then the two-sided integral will converge in the strip Ue1< Re 8< U.2 (10-18) providedthat U.1< Uc2;otherwisetherewouldbenooverlapofthetwo half planes.Equation (10-18)suggests that two abscissas of convergence (Udand U.2)aredefined for the two-sided Laplace integral. Allthediscussionsofthesingle-sidedLaplaceintegralgiveninSees. 10-3 through 10-6 apply to the second integral on the right of Eq. (10-17). It isnecessaryonlyto consider I( - t)and to changethe signof8.The abscissa of convergence U"zcan be 'determined by the conditions of either 296COMPLEXVARIABLESANDTHELAPLACETRANSFORM Theorem 10-1or Theorem 10-2.Theorem 10-1applies if f( -t) is EO,ao, where then CT c 2equals-ao.Theorem10-2applies if fo"f( -t)e·.. 1 dt exists,andthen CT c 2willbetheleastupperboundofadmissiblevalues ofRe S02.Thus, CT c 2can be found fromproperties off( -t), and then the range of convergence given by relation(10-18)is established. Some f(t)functions will yield strips of convergence of finitewidth,and others willnot.The function { eal f(t)=(/'1 0< t t U g The condition Re 8> U g is important in the sense that this condition can always be satisfied.The abscissa of convergenceofthe Laplace integral ofthe sum oftwo functionswillbe the larger of the two abscissas ofthe individualfunctions.Thus,if.c[f(t)]and.c[g(t)]eachexist,then .c[f(t)+ get)]alsoexists and isgiven by .c[f(t)+ get)]= .c[f(t)]+ .c[g(t)](10-25) Thesituationissomewhatdifferentfortwo-sidedtransforms.It is merelyadetailtoshowthatEq.(10-24)canbemodifiedfor.cM(t)], giving (10-26) but there isnogeneraltheorem foradditionoftwo-sided Laplacetrans- forIns.The reason can be foundby considering the formal expression 1-". f(t)e- d dt+ 1-....g(t)e- ot dt=1-....[f(t)+ g(t)]e- ot dt Even though each of the integrals on the left might converge,the integral on the right exists(and an equality signcan beused)only if there isan overlap of the strips of convergence ofthe integrals on the left.Thus, if the two-sided Laplace integrals of f(t)and get)have overlapping strips of convergence,it is true that f(t)+ get)has atwo-sided Laplace transform given by .cM(t)+ get)]=.cM(t)]+ .c 2 [g(t)] If thereisnooverlappingstripofconvergence,theabovesumonthe right ~exist but the left side will not exist.The right side will be the two-sidedtransformforsomefunctionotherthan f(t)+ get).Further explanation ofthis statement is found in Secs.10-16and 10-17. 10-11.Laplace TransformsofSome TypicalFunctions.Later onwe shalldevelopsomegeneralpropertiesofLaplacetransforms.Forthe present, some feeling for these functions will be obtained by working out a fewexamples. 1.f(t)=rf".Convergence is forRe 8>a.Weshall followthe plan ofSec.10-9,first integrating the defining integral with 8real,as follows: r" e-(I1-;all dt=_e-(I1-o>,I"=_1_ )0U- a0u- a Replacing (Iby 8gives THELAPLACETRANSFORM £(eo l )=_1_ 8-a 301 (10-27) 2.f(t)= 1.ConvergenceisforRe 8> 0,andthiscasecanbe obtained fromthe aboveby taking a= 0,giving 1 £(1)=- 8 (10-28) 3.f(t)= sinbt,andcosbt.ConvergenceisforRe 8> 0.Again using 8real,weget the following: 10'"'(sinbt)e-at dt=e-O't( -(I s ! ~~~b cosbt)I: b = (12+ b 2 10'"'(cosbt)e-at dt=e-O't( -(1 c:: ~t b sin bt)I: (1 = (12+ b 2 Againreplacing (1by 8,wehavethe results £(sin bt)=82! b 2 8 £(cos bt)=S2+ b 2 (10-29) (10-30) 4.f(t)=eo ' sin bt, and eo ' cos bt.This is a special case of a general case to be treated later, where at function,having aknown transform F(s), is multipliedbyanexponential.Fortheabovethreefunctions,conver- genceisforRe s> a.Therequiredintegrationisreadilyperformed, being basically the same as case 3, but with-(1 replaced by a- (I.Thus, referringto that case,wehave ('"'(sinbt)e(G-O')1dt= e(G-O')I[(a- (1)sin bt- b cosbtlI'"' Jo«(1- a)2+ b ' 0 b «(I- a)2+ b 2 ('"'(cosbt)e(a-a)l dt=e(G-O')I[(a- (1)cosbt+ b sin btl/ .. }o«(I- a)2+ b 2 0 (I-a =«(I- a)2+ b 2 302COMPLEXVARIABLESANDTHELAPLACETRANSFORM In each case the results are dependent upon the factthat IT>a.Now IT is replacedby s,to yield .e{e'"sin bt) b (s- a)2+ b 2 s-a .e{e'"cosbt)=(s_a)2+ b2 (10-31) (10-32) 5.f{t)=tn,Where n Is an Integer.ConvergenceisforRes>0.The defining formulacan be integrated by parts,giving 10 ..tn., ,..10" t"e--f7'dt=- ~+ ~tn-1e-·' dt oIT0IT0 Theintegralontherightexists,andthelowerlimitcanbeusedinthe firstterm,ifn~1.SinceIT>0,theexponentinthefirsttermgoes to zeroas tgoesto infinity.Thus wehavethe formula .e(tn)=?!:.e(t n - 1 )(10-33) s Case2isthesameasthepresentcase,ifn=O.Thus,Eq.(10-33) leadsby induction to the sequence .e{tO)= .! s 1 .e(t)=82 .e{t 2 )=~ Sl (10-34) 6.JCt)=1/0.This functionis APCby virtue ofitsbehaviornear thesingularityat t =0.However,it isalsoEO,O,andsoitsLaplace integral converges forRe s>0.Here wehave acase toward which the comments ofSec.10-9weredirected.Aswe shall see, it is essential that s shallbe replaced by IT.In the integral replace ITtby X2,giving (10-35) THELAPLACETRANSFORM303 This integral iswellknown and can be foundin tables ofimproper inte- grals.Inwritingvi andv-;i intheaboveformulasweunderstanda singlebranchofthe or functions.That isto say,aminussign is not admitted.Now,when weanalytically continue 0, we go into only one sheet of the Riemann surface of the function the sheet in which the points ofvu are located.Let us then usethe symbolVs to imply this single-valuedbranchofthefunctionAccordingly,theresultis (10-36) By firstconsidering the Laplace integral for the real number 0"we are able to arriveat the correct choiceofthe twopossiblevalues of If thiscasehad been treated throughout with the variable 8retained, the formal manipulations of variable change would have led to the factor l/Vs.However,wewouldnotthenhaveaclearmeaningforVB. Notation is merely a symbolism for ideas; symbols which are not properly defined have nomeaning,apoint which is illustrated by this example. 7.f(t)=Vi!',k1andOddInteger.Thisfunctiongivesconver- genceforRe s> O.Asin case5,integrationbyparts yieldsageneral recurrence formula,as follows: r" Vi!' e- IT1 dt=_Vi!' e- IT1 /"+ r" e- IT1 dt Jo0"020"Jo If k1,the lower limit can be used in the first term on the right and the integral exists.The result canbe stated as k1 and odd (10-37) . In similarity with Eqs.(10-34), this yields a sequence of formulas, starting withvt-l, which is obtained fromcase6.Thus, 304COMPLEXVARIABLESANDTHELAPLACETRlI.NSFORM In this general formula the root of apower of s is always interpreted to be on that sheet of the Riemann surface on which are found the values of y?+i. 8.f(t)=PulseofUnitHeightandDurationT.Thisfunctionis definedas f(t)={ ~ and it has the Laplace transform 0< tTo,where Toisany positivenumber,then F(s)isan entirefunction. PROOF.InthiscasetheLaplaceintegralreducestoan integral with finitelimits,namely, (To F(s)=10J(t)e- BI dt SincethelimitsarefiniteandtheAPCconditionensuresintegrability, thisintegralexistsforallfinitesandwemaysaythatitsabscissaof convergenceis- ~ .FromSec.10-8 weknow that the Laplace integral is regular to the right of the axis of convergence, in the finite plane in this case. If youwillrefertoTable10-1,youwillfindillustrationsofTheorems 10-3 through 10-6.Many more general properties of F(s) can be derived. Someofthesimpleronesaregiveninthenext section,andothers are developedinlaterchapters.Theintentionhereistopresentonlythe relatively simple properties that canbeproved relatively easily. 10-13.The Shifting Theorems.Supposethat wehavethetransform F(s)ofatransformablefunctionJ(t)andthenconsiderthetransform ofthe function whereaisrealor complex.If O'cistheabscissaofconvergencefor J(t), then the integral converges for Res> O'c- Rea.However, this integral is an expression forF(s + a),showingthat,if theLaplacetransformisknownforany function,the transform of that function multiplied by an exponential can immediatelybeobtainedbyasimplechangeinvariable.Thereisa 310COMPLEXVARIABLESANDTHE- LAPLACETRANSFORM "shift,"ortranslation,inthesvariable.Thuswehavethegeneral result where .£[e-alf(t)]=F(s+ a) F(s)=.£[f(t)] (10-43) For thenextcaseagainassumethat thefunctionf(t)has a transform F(s),andconsiderashiftinthetvariable.From f(t)anewfunction is found by changing the variable to t- T, where Tis a positive constant, whilestipulatingthatthenewfunctionshallbezerofort< T.With the aid ofthenotation forthe unit step, u(t)={ ~ this functioncanbewritten t< 0 t> 0 f(t- T)u(t- T) (10-44) Inthegraphicalinterpretation,thistransformationshiftsthegraph anamountTinthepositivetdirection.Thetransformisrepresented by the integral fo 00f(t- T)u(t- T)e-'! dt=fT"f(t- T)e-&t dt The factoru(t- T)isdroppedinthesecondintegral becausethelower limithasbeenchangedtoT.Nowchangethevariableofintegration to t'=t - T,giving .£[f(t- T)u(t- T)]=e-· T fo"f(t')e-' " dt' = e-.TF(s) These resultsaresummarized inthe followingtwo theorems: (10-45) Theorem10-7.If f(t)has aLaplacetransform F(s),then theLaplace transform ofe-a'f(t)isF(s+ a),wherea isreal or complex. Theorem10-8.If f(t)has aLaplacetransform F(s),then theLaplace transform of f(t- T)u(t- T)ise-·TF(s),whereTis realand positive. Examples of applications of these two theorems are found in Table 10-1. To cite one,we can findthe transform of cosbt by knowing the transform of f(t)=1,whichisl/s.Thus, efbl+ fr/'b' cosbt=2 1(11)8 .£(C08bt)=28- jb+ 8+ jb= 82+ b2 THEL A P k ~ C ETRANSFORM 311 10-14.LaplaceTransformoftheDerivativeoff(t).Weshallnow obtainarelationshipbetweentheLaplacetransformofthederivative ofafunctionandtheLaplacetransformofthefunctionitself.We assume that f(t)is continuous fort> 0 and has alimit at t=0 and that its derivative is APC.We also require/(t)to be ofexponential order ao. Consider the Laplace integral,written with 8=U,as follows: r" f'(t)e-vl dt=lim(A!'(t)e- v1 dt JoA_ ..JO Theintegralontherightmeetstheconditionsforintegratingbyparts (Theorem 8-4),namely,continuity of f(t)and e- V1 .Thus,weget (A !'(t)e- v1 dt=limf(t)e-V1I A + u(Af(t)e- v1 dt Jo.-0'Jo = f(A)e- vA - f(O+)+ ufo Af(t)e-v1 dt(10-46) It isnecessarytowrite lim feE)=f(O+) .-0 inorderto saypreciselywhatwemean,becausequiteoften f(t)isdis- continuous at t=0and f(O)may beundefinedordefinedassome value otherthanf(O+),usuallyf(O+ )/2.Thisistheonlydiscontinuity permitted in thetheorem wearedeveloping.A similartheorem,where f(t)is allowedother points ofdiscontinuity, isconsidered inChap.12. Werecall that f(t)isEO,ao.Therefore,by relation(10-5), when u>ao.Also, limf(t)e-vA=0 A ....... lim( Af(t)e-vl dt A--+oo}o convergestoF(u) ,foru> ao,byTheorem10-1.Thus,for(I> ao, Eq>(10-46)yields or fo"f'(t)e-v1 dt=uF(u)- f(O+) £[f'(t))= sF(8)- f(O+)(10-47) ThisproofshowsthattheabscissaofconvergencefortheLaplace integral of the derivative function is at most no larger than ao.We also note that f(t)must be of exponential order but that!'(t) is not necessarily of exponential order.For example, f(t)= cose,l 312COMPLEXVARIABLESANDTHELAPLACETRANSFORM isEO,O,but its derivative f'(t)=-2te 'l sin e,l isnot of exponentialorder.Butby theproofjust completedweknow that it hasaLaplaceintegra.lwithzeroforits abscissaofconvergence. Thisproofcanbeappliedrepeatedlytosuccessivederivatives,as longasthenext-to-the-la8tderivativeisEO,ao.Theresultisstated as follows: Theorem 10-9.Letf(t) and its derivatives of orders up to and including ordern- 1becontinuousfort>0,withlimitsexistingat t=0,and of exponential order.Then the Laplace transform of f',,)(t)is £[j lTl.Thus,thefunction 314COMPLEXVARIABLESANDTHELAPLACETRANSFORM hasaLaplaceintegralwhichconvergesforRe 8> 0"1andaLaplace transform givenby It is of interest to observe how the abscissa ofconvergence 0"1is related toO"c,theabscissaofconvergenceoff(t).Wefoundthatifo"c< 0 then0"1=0,butif o"c> 0then0"1=O"c.Toillustrate,considerthe example f(t)=e-bt b>O Its abscissa ofconvergenceis-b, but the integral ofthis function (t1- e- bt }oe- lYr dT=b has zero as its abscissa of convergence.Wehave now proved the follow- ing theorem: Theorem10-10.If f(t)isAPCand hasaLaplacetransform,then the functionJot f(T)dThas aLaplacetransform given by (10-52) Asan exampleofan applicationofthis theorem,wecan get .c(sinbt) from .c(cosbt)by recognizingthat (tsin bt }ocosbx dx=-b- From Theorem10-10 weimmediately have .c(sin bt)=_1_ b8 2 + b 2 .c(sinbt)=82! b 2 A theorem similar to Theorem 10-10,but in whichf(t) is of exponential order,has amuch simpler proof.Its proof isleft to you asan exercise. 10-16.lnitial- andFinal-valueTheorems.Thederivationofthe formula for the transform of a derivative provides two theorems which are sometimesusefulinanalysis.Letf(t)becontinuousfort> 0,witha limit at t= 0,ofexponentialorder,andwithaderivative f'et)which is APC.Thesearethe conditions forTheorem10-9,and accordinglythe transform of f'(t)exists,and Eq.(10-47)applies.By Theorem10-4 lim.£[f'(t)]=0 -- THELAPLACETRANSFORM and therefore it followsfromEq.(10-47)that limuF(u}= f(O+) "...... .. 315 (10-53) We shall now see that the value approached by f(t)as t becomes infinite canalsobedeterminedfromF(s).Assumethesameconditionsas beforeonf(t)and f'(t),with the additional stipulation that theLaplace integral fo" !'(t)e-"dt shall converge for s= O.From Theorem 10-2 it follows that convergence of this integral is uniform fors real and nonnegative,and sowecan take the limit as u-+ 0 insidethe integral,asfollows: lim{ .. !'(t)e- ff 'dt=( .. I'(t) dt=f( co)- f(O+) "......01010 Now ifyou willreferto the equation precedingEq.(10-47)and take the limit indicated above,theresult is f( co)- f(O+)=lim uF(u)- f(O+) ~ O or lim uF(u)=f( co)(10-54) "......0 These resultsarederivedhere forarestricted class of functions.The requirement that f(t)shallbecontinuous wouldperhaps seem to provide an unreasonable limitation.In Chap.12weshall review this topic again andshallfindthatsomeoftherestrictionsatpresentplacedonf(t) can beremoved.However,the results givenaboveare as general as we areabletoprovewiththetheorypresenteduptothispoint.Pending the treatment of the more general conditions, a formal statement of these resultsisomittedhere.It shouldbementioned,however,thatEqs. (10-53)and(10-54)statetheresultsoftwotheoremswhichareknown, respectively,asthe initial-value theoremand the final-valuetheorem. 10-17.NonuniquenessofFunctionPairsfortheTwo-sidedLaplace Transform.In Sec.10-2it isshownthatthetwo-sidedLaplacetrans- formcan bedefinedasthe sum oftwoone-sidedtransforms.To reiter- ate,if5'(s)= .eM(t»),F1(S)=.e[f(t»),and F 2 (s)=.e[f( -t)], then (10-55) Inthiswayweseethatatableofone-sidedtransformscanalsobe usedtoobtaintwo-sidedtransforms.Butthetablecannotbeusedto obtainf(t)if5'(s)isknown.Inordertodeterminef(t),knowledgeof 316COMPLEXVARIABLESANDTHELAPLACETRANSFORM 5'(S)must besupplemented with directionsas to how it istobesplit up into FI(s)and F 2 (-S), or someequivalent mformation. In order to clarify these ideas,assume that two functions fo(t)and fIlet) are given,with converging single-sidedLaplace integrals as follows: and FuCs)= fo"fo(t)e- d dt F,,( -s)=fo"f,,( -t)e"dt Res> iTo Res-2 Rea 0andgives zerofort< O.In appraising these com- ments werecognizethe role played by the FIG.10-10. The Br contour whichBrcontour.Theimportantfactisthat yields zero jCt)fornega.tivet. fromthelocationofsingularitiesofthe given F{s) weknow Br must be to the right of all singular points, and there- foreEq.(1O-60)gives a unique functionf{t)u(t). ThefactthatFes)doesuniquelydeterminef{t)u{t)meansthatwe canadoptanotationtoindicatethatrelationship.Thenotation F{s)=.e[f(t»)hasbeenintroducedtosymbolizethatf{t)determines F{s)uniquely.Nowwearegoingintheotherdirection,andsoitis convenient to usethe notation f{t)u(t)=.e-1[F{s»)(10-61) torepresentEq.(10-60).Thefunction.e-1[F(s»)isoftencalledthe inversetransformofF(s). Inthetwo-sidedtransform,5'(8)isuniquelyrelatedtof{t)bythe formula 5'(s)=1-....f(t)e-"dt andthisfactisimpliedwhenwewrite5'(8)= .eM(t»),but,aspointed THELAPLACETRANSFORM321 out earlier, a notation similar to Eq.(10-61)cannot be employed, because oflack of uniqueness. ThenumericalexamplegiveninSec.10-17providesacaseinpoint. There weobtained t>O)11 t-1, and whereconvergenceisfor- 2< Re s< -1.Totheseweaddathird case, whereconvergenceisforRe sJR ~ "C (10-63) The detailsof this variable change aregiveninFig.10-12. Jordan's lemma applies to the integral ofEq.(10-63)fortwo cases,as follows:If limF(z+ 0'1)=0 I z l ~ " uniformly for- i ~8~~(10-64) andt< 0,thentheintegralover0 1 approacheszeroasRapproaches infinity.Similarly,if limF(z+ 0'1)=0 I z l ~ " uniformly for ~~8~3;(10-65) andt> 0,thentheintegraloverO 2 approacheszeroasRapproaches infinity.Theseconditionsmust. be satisfied,respectively,fort< 0and t > O.In many practical cases F(s)meets the stronger condition limF(s)= 0 I · I ~ · uniformly forallq,(10-66) whichofcourseissufficientforbothconditions(10-64)and(10-65). 324COMPLEXVARIABLESANDTHELAPLACETRANSFORM Wehavereachedanimportantmilepost.If conditions(10-64)and (10-65)aresatisfied,or ifcondition(10-66)issatisfied,wecanevaluate the integral ofEq.(10-63)or its equivalent, f(t)u(t)=21.(F(s)e"ds 7rJJBr (10-67) byclosingthepathontherightorleft,depending,respectively,on whether t< 0 or t> o. *Then the calculus of residues comes into play. If F(s)issingle-valued,whent< 0theintegral is equal to-27rj multi- pliedbythesumoftheresiduesinthehalfplanetotherightofBr, and whent>0the integral is27rjmultipliedby the sum ofthe residues in the half plane to the left of Br.Of course, weknow that F(s) is regular to the right of Br, sinceBr isto the right of the axisof convergence,and soweget zerowhen t< O.Therefore,Eq.(10-67)reduces to f(t)u(t)={ ~ u mof residues to left ofBr t< 0 t>O (10-68) If F(s)is multivalued, the paths C 1 and C 2 must of course be appropri- atelymodifiedtoavoidcrossingbranchcuts,insomemannersuchas that described in Sec. 8-12. Thetwo-sidedtransformff'(s)isenoughdifferenttowarrantsum- marizingresultsinthiscase.If ff'(s)satisfiesconditions(10-64)and (10-65), or condition (10-66), and if ff'(s)is single-valued, then the calculus ofresidues is used again.However,nowpath Br liesinavertical strip, with singular points of ff'(s)to the right of this strip as well as to the left. Therefore, a nonzero result is obtained when t< 0, and we can summarize: f(t)_{- sum of residues to right of Br - sum ofresidues to left of Br t0 (10-69) In thisequationwehaveconfirmationofthe statement madeinSec. 10-17,that f(t)isnotuniquelydeterIninedifff'(s)isknown.Thereare singularitiestotherightandleftofBr,andBrcanbemovedintoa different strip of regularity, thereby changing one or more poles from one sideofBr to the other. 10-20.Evaluating the Residues(The Heaviside Expansion Theorem). On the basis of the previous section, we can now derive one of the classical theoremsinLaplacetransformtheory.Thisresultfirstappearedin engineering literature in the operational calculus of Oliver Heaviside. InEqs.(10-68)and(10-69)wearedirectedtofindresiduesofthe function F(s)e T. THELAPLACETRANSFORM325 arational function(ratioofpolynomials)withthedegreeofthe denom- inatorat leastonegreaterthan thenumerator.Thiscondition ensures that condition (10-66)will be satisfied.F(s) could be one of the functions tabulated in Table10-1,for example. In Chap.5weconsideredthe representation ofaratio ofpolynomials inapartial-fractionexpansion.Therewillbepolesat 81,S2,•••,SM oforders N 1,N 2,•••,NM.The expansionwilllook like (10-70) Allweneed do,then,isto findtheresidueforthe typical term an,ke lt (s- s/;) .. This term itself has a Laurent expansion, for which we want the coefficient of(s- Sk)-1,theresidue.Thisproblem isworkedout in Chap.5,and the answer isgivenbyEq.(5-80)as Residue=".(e d ) akd n - 1 I (n- 1)! ds n - 1 ._ •• (10-71) Thus,whenF(s)isarationalfunction,itsinversetransformwillbe asum ofterms likeEq.(10-71).Note that there isacertain amount of workinarrivingat thesetermswhichisnot shownhere,the findingof. thecoefficientsa.. ,kfromthepartial-fractionexpansionofF(s).The combination of Eqs.(10-70)and (10-71)is an expression of the Heaviside expansiontheorem. Although Eq.(10-71)represents all possible cases when F(s) is rational, thereisanimportantsubclassificationwhenSiciscomplex.It canbe shownthatforphysicalproblemsF(s)isreal.Assumethatthisisthe case,andrecall,fromChap.7,thattheremustbeaconjugatepoleat Sk+1=Sit.The coefficientsinLaurent expansionsabout these twopoles willbeconjugat.es,giving a.. ,It+l= an,lt (10-72) Consequently,thepartial-fraction expansionof F(B)willincludethe two terms 326COMPLEXVARIABLESANDTHELAPLACETRANSFORM In simiiarity withEq.(10-71)weget Sf 'dt .. - 1 (-.1- "') urn0tworeSIues=(n_I)!a ... k e + a...k e • This isreduced to amoreusefulformby writing with the result a... k =A ... kei"·" Sk= 0'1' dw (11-7) The formulaforw(t)inEq.(11-4)can beput intoamoreconvenient formbychangingthevariableofintegrationfromTto-T,withthe result w(t)=f: ..f(T)g(t- T)dT (11-8) TheintegralinEq.(11-8)iscalledaconvolutionintegral.Lateron othertypesofconvolution integralswillbedefined.Wenow state this result formallyasthe followingtheorem: Theorem11-1.Letf(t)andg(t)eachbeAPC,whileoneofthemis bounded asIt I becomes infinite and has at most afinitenumber of infinite discontinuities,andletbothbeabsolutelyintegrablefrom- 00to00, with respectiveFourier transforms 5'(jw)andg(jw).Then, w(t)=f-....f(T)g(t- T)dT is an APC functionof t,and is continuous if either f(t)or g(t)isPC, and its Fourier integralconvergesto 5'(jw)g(jw) 11-3.ConvolutioninthetPlane(Two-sidedLaplaceTransform). Let f(t)and g(t)be APC functions for which the integrals f _....If(t) le- ol dtand f-....Ig(t)le-"dt have strips of convergencewhichoverlap in astrip designatedby ITa< Re8< ITb Also assume that there is a number ITlbetween ITaand ITbsuch that Ig(t) le-ao. Thefollowingcommentsareofferedinappraisalofthistheorem. Comparison showsthat Theorems 12-1and 12-2both deal with behavior as s approachesinfinity along aradial line,making an angleless than or equalto7r/2withtherealaxis.Therearetwoimportantdifferences, however.In Theorem12-1thelinecanradiatefromanypointin the 8plane,and F(s)approaches zero u m f o r m ~ ywith respect to its angle in a righthalfplane.InTheorem12-2F(s)approacheszeroinasimilar sector,but the apex must bein the regionofconvergenceoftheLaplace integral;and there isnoproof that zero is approached uniformly.Thus, FURTHERPROPERTIESOFTHELAPLACETRANSFORM357 Theorem12-2isnotstrongenoughtoensureapplicabilityofJordan's lemma to theinversion formula. 12-3.FunctionsofExponentialType.YouwillrecallfromChap.5 thatanentirefunctionisananalyticfunctionhavingnosingularities in the finiteplane.AnyTaylor series of such afunctionhas aninfinite radiusofconvergence.Lettbetherealpartofacomplexvariable w= t + ju, and let few)be an entire function.In addition, assume that it is possible to findnumbers M and 'Ysuch that for all w If(w) I Me'YI1D1 (12-1) whereMand 'Yarerealnumbers greater than zero.Many functionsfit intothiscategory;functionssuchassinbt,e al ,anypolynomialin tare typical.Anentirefunctionsatisfyingcondition(12-1)issaidtobeof exponentialtype.Thisdesignationisnottobeconfusedwithexpo- nentialorder.Functions ofexponential order arenot necessarily entire functions and, indeed,need not even be defined forcomplex values of the variable.We note that the function e- I 'isof exponential order,but not ofexponentialtype,becausee-(iu)'=e u 'cannotbedominatedbythe exponential inEq.(12-1).Also,any functionhaving discontinuities,or discontinuousderivatives, cannot beof exponential type,becausesuch a functionisnot analytic. In allbut the trivial case f(t)==0,the set of values forwhich relation (12-1)is satisfied will have agreatest lower bound 'Yowhich can never be negative.Thisnumber mayor maynotbeamemberofthe setof'Y's forwhichrelation(12-1)is true.Weshall call 'Yothe orderof f(t). Since few)is an entire function,it possesses all derivatives and weshall nowshowthat eachderivative isofexponential type,ofthe sa.meorder 'Yoas f(t).TheCauchyintegral formulaforthe nth derivative is j(w)=n!.rfez)dz 27rJ}c (z- w)n+1 (12-2) where C is a circle of radius R centered at z=w.The change of variable z=w+ Re i6 dz= jRe i 'dO leadsto j'Yo 358COMPLEXVARIABLESANDTHELAPLACETRANSFORM Thequantityinparenthesesisindependentofw,andsoweconclude that f''')(w)isofexponential typeand oforder 'Yo.Amoreconvenient formfortheabove,forlateruse,isobtainedbychoosingR=n/'Y, which can be donebecauseR, the radius ofthe circleused in the Cauchy integral formula,isarbitrary.Thus, If(")(w) I < Mn!e"'Y"e'Y/1D/'Y> 'Yo(12-3) n" Next we show that a function of exponential type, and all its derivatives, areofexponentialorder.This can easilybe shown by observing that if If(w)1~Me'Y/1D/ If(")(w)1~M"e'Y/ 1D / then for w=t> 0(wreal)wehave If(t)le-'~Me-(a-'Y)' 1f''')(t)le-'~M"e-(a-'Y}/ (12-4) For each of the above, t can be made sufficiently largeto make the right- hand sidearbitrarily small,if the condition a>'Y issatisfied.Thus,f(t)and j'Y(12-8) n ~Bn=rsr Theconditionlsi>'Yissatisfied when u>'Y,and therefore this limit is lessthan1.Wehavejust completedthe ratio test forthe series Bo+ Bl+... showingthat convergenceoccursforu>'Y.But aseriescan converge onlyif thetermsapproachzero,andsinceIA"I< B .. ,wehaveproved limIA .. I =0ifu>'Y (12-9) n-+oo It followsthat Eq.(12-5)can bereplacedby the series u>'Y If apowerseriesinpositivepowersofl/s convergesat somepointl/er, it willconvergeinthe circular region l . ~ ~ < l lsi- u'Y and thereforethe aboveseriesconverges intheregion lsi> 'Y>'Yo It issufficientto designatethisregionby lsi>'Yo (12-10) Thus,althoughtheLaplaceintegralhasahalfplaneofconvergence, theseriesexpressionforthetransformconvergesintheregiondefined aboveand accordinglyweget the result 181> 'Yo (12-11) 360COMPLEXVARIABLESAND.THELAPLACETRANSFORM Next weobservethat theTaylor expansion of f(t)is 00 f(t)=\' j N Vla,,1- 'Yo< E orla,,1< ('Yo+ E)" (12-15) since 'Yo> O.For each n;:::;;Nwe can define a number M" by the relation la .. 1=M .. ('Yo)" TherewillbeonevalueofM ..whichisthelargestofthisfinitenum- berofterms;LetMbethislargestM ..orunity,whicheverislarger. Accordingly, 362COMPLEXVARIABLESANDTHELAPLACETRANSFORM Relation(12-15)isalsotrueif Misincludedasafactorontheright, becauseM!i;;1.Thus,wehave la,,1< M('Yo+ E)"all n (12-16) ...... and 12: t" I2: I:i Iltl"< ML[('Yo n-O,,-0n=O Buttheseriesontherightistheexponentialfunctionofargument ('Yo+ E)ltl,which converges forall t.Therefore, the series inEq.(12-14) alsoconverges,andhencef(t)isanentirefunction.Also,theabove can bewritten f(t)< Me 'Yo,then .c-1[F(s)] is of exponential type and there is a number Msuch that If(t) I < M e'Yi1iall t where'Y>'Yo· 12-4.A Special Class of Piecewise Continuous Functions.Many func- tionsofimportanceinengineeringaremadeupofcontinuoussections, each ofwhichisaportionofafunctionconsisting ofsums oftermslike t"et".The followingtwoexamples illustrate this type of function: Case a 1Oa, Now,if (1~a'>a,the above gives n> N' SinceEwasarbitrarilychosen,lisalsoarbitrary andsoitfollowsthat the series converges uniformly for Re 8~a'>ao.Absolute convergence showsthattherearrangementsuggestedfortheseriesinEq.(12-20)is valid,and accordinglyweassumethat f(t)isEO,aoand write 00 /0"f(t)e- vt dt= Ao- 1 (Ak- Bk_1)e- vT • k-l (1> ao(12-24) Eachintegralinl!.:qs.(12-23)hasahalfplaneofconvergence,andso wecan regardAkandBkeach asfunctionsof8whicharein reality the respectiveLaplacetransformsof !k(t + T k )and /k(t+ T k + 1 ).Accord- ingly,let us define k=O k ~ O (12-25) 366COMPLEXVARIABLESANDTHELAPLACETRANSFORM giving,finally, .. F(s)=1 ",,,(S)-IT. 1:-0 Res>ao(12-26) Thecasewherethenumberoftransitionpointsisafinitenumber N, represented by Eq.(12-21),is simpler.In that case it isunnecessary to beconcernedabout convergenceofaseries.Theonly formalchange is in the summation limit,and wehave N F(s)=I "'k(S)C· T • k-O (12-27) Equations (12-26)and (12-27)have aunique form.AB expected from theearlierexample,eachtransitionpointproducesafactorlikee- IT •• Furthermore,each",,,(s)istheLaplacetransformofacombinationof terms liketne al and isthereforeknownto bearational functionin s. The casewhereeach sectionof J(t)isaconstant, fleet)=D" is ofparticular interest.From Eq.(12-25)wethen have Do "'o(s)=- s "',,(s)=D"- D"_lk~1 s and Eq.(12-26)gives .. F() Do+ 2: D"- D"_lT S=- e-" ss "-1 (12-28) You should beawarein this development ofthe importance ofhaving f(t)ofexponentialorder,inadditiontobeingPCSEPintype.The latter condition does not ensure the former.For example, the successive fleet)functionscanhavemultiplyingconstantsthatincreasesorapidly with increasing kthat J(t)might not be of exponential order. ABatheorem,wecan now state theseresultsas follows: Theorem12-5.Let J(t) be made up of sections of functions of combina- tions of terms like t"ea ' , and let Jet)be EO,ao.Furthermore,letT"repre- sentthepointwheref(t)experiencesatransitionfromsectionf"-l(t) tosectionJ,,(t),takingk=0astheorigin.Then, J(t)hasaLaplace transform in the form .. F(s)=I ",,,(s)e-· T • k-O Re s> ao FURTHERPROPERTIESOFTHELAPLACETRANSFORM367 whereeach ",,,(8)isarational functiongivenby ",,,(8)={ .c[fo(t)] .c[f,,(t+ T,,)- /"_l(t+ T,,)] k=O k~1 The series expression forF(8)convergesuniformlyforRe8~a'> ao. Corollary.H section k of a function meeting the conditions of Theorem 12-5 isaconstantD",then the ",,,(8)functionsin that theorem are given by k~1 Intheabovetheoremwedonot haveanyparticularrequirementon the spacing of the transition pointsT k •Ofcourse,the results presented here include the case where the spacing is uniform and where each section is a repetition of the previous one; i.e., the periodic functions are included. However,this theorem gives somewhat lessthan the maximum informa- tion about the transform of aperiodic function.A detailed consideration of the periodiccaseisgiven in Chap.15. 12-6.LaplaceTransform of the Derivative of aPiecewiseContinuous FunctionofExponentialOrder.InSec.10-14,whenconsideringthe Laplacetransformofthederivativeofafunction,weconsideredthe functionto be continuous.This isin agreement with the usual practice of requiring a function to be continuous if we are to discuss its derivative. Of course, situations like point T 1in Fig.12-2 are admitted, where right- andleft-handedderivativesexistatthepoint,makingthederivative approach differentvaluesfromthe two sides. Now suppose that f(t)has isolated points of discontinuity, likeT2and Ta.Noderivativecanbedefinedatthesepoints,butthefunction canbedifferentiatedat allneighboringpoints.PointsTl andT2differ to the extent that the derivative(in the one-sided sense)isdouble-valued at Tl and does not exist at T 2 •Thus, it is not unreasonable to talk about the"derivative,"whenafunctionhasisolateddiscontinuities,ifwe under8tandthatthederivativefunctioni8undefinedatthepoints0/di8- continuity.In this sense weshall definethe derivative ofaPC function and shall usethe symbol !'(t)to represent it. Having defined f'(t)in themannerdescribedabove,wenow formthe function (12-29) This function is not affected by the fact that!'(t) is undefined at isolated points, because these points form a set of measure 0 and,by Theorem 8-5, 368COMPLEXVARIABLESANDTHELAPLACETRANSFORM we understand that the value of an integral is not affected if the integrand is undefined over such aset.An integral isacontinuous functionofits upperlimit.Thusfo(t)iscontinuousandhasthesamederivativeas f(t)atpointswheref'(t)isdefined.Figure12-2showsanexampleof foO)and alsothe staircase function f(t)- fo{t). r- ,--------------------- -------r--------- II(e) I L___________ _ FiG.12-2.Resolutionof adiscontinuousfunctionintocontinuous and discontinuous components. Thefactthatf(t)isofexponentialorderdoesnotensurethatfo(t), and also f(t)- fo(t),willbeof exponential order.An illustration can be derivedfromthe function e,l which isnot ofexponential order.At values t..=y'log nninteger thefunctionhasthevaluen,andsoif adownwardjumpofunityis introducedateacht.. ,afunctionlikeFig.12-3(whichisnevergreater than1)willbeobtained.If Fig.12-3islabeled f(t),thenEq.(12-29) will yield fo(t)=e,l- 1 Whereas f(t)isofexponential order 0,fo{t)isnot of exponential order. FURTHERPROPERTIESOFTHELAPLACETRANSFORM369 1.0 0.20.40.60.81.01.21.4 t--- FIG.12-3.Example of afunctionof exponential order forwhichfot I'(T)dTis not of exponential order. In viewofthe abovecomments,it isnecessaryalsotorequire fo(t)to beofexponential order;andthen f(t)- fo(t)willalsobeofexponential order.WecanadoptthenotationofthecorollaryofTheorem12-5 and write f(t)- fo(t)=Dk(12-30) andthisfunctioncanalsobewrittenasaseriesofdisplacedsteps,as illustrated inFig.12-2e,giving where ... f(t)- fo(t)=I BkU(t- T k ) k-O Bo=Do Bk=Dk- Dk- 1k~1 (12-31) (12-32) ThecorollaryofTheorem12-5isapplicabletoEq.(12-30),and,using the notation ofEq.(12-32), (12-33) Note that this result would have been obtained by a formalterm-by-term transformationoftheseriesinEq.(12-31).However,Theorem12-5 provides justification ofthis step. . Equation(12-29)fitstheconditionsofTheorem10-10,andtherefore .e[fo(t»)=.e[f'(t») s (12-34) Asusual,let F(s)=.e[f(O].Then Eq.(12-33)becomes ... .e[f'(t)]=sF(s)- L Bke-· T • k-O (12-35) ThisresultisageneralizationofTheorem10-9.Inthatearlier theorem,fCt)iscontinuous.Thus,thepresentresultshouldreduce 370COMPLEXVARIABLESANDTHELAPLACETRANSFORM to Theorem 10-9 if we set B"=0, except Bo= J(O+ ).Then Eq.(12-35) gives £[f/(t)]...sF(s)- Bo inagreementwithTheorem10-9.ThegeneralizationimpliedinEq. (12-35)isnot evident fromthe earlier theorembecausetheexponential eOis not in evidence on the term J(O+).We can now state the following moregeneral theorem,ofwhich Theorem10-9 is aspecial case: Theorem12-6.Let J(t)be apiecewise continuous function of exponen- tial order,and let the discontinuous function .. J(t)- Jotf'(T)dT=lBkU(t- Tk ) k-O alsobeof exponential orderao,it being understood thatJ'(t) is undefined at pointswhere J(t) is discontinuous.Then, if F(s)=£[J(t)], the Laplace transform of f'(t)exists and isgivenby .. £[f'(t)]=sF(s)- 2:Bke- or •Res>ao 10=0 Equation(12-35)can alsobe written .. F(s)= £[I'(t)] + \' Bke-or•(12-36) sLts k=O whichbearsacertainresemblancetoTheorem12-5.However,inEq. (12-36)weare told that exponential terms like Bke-or• s must appear,but fromthis equation weknow nothing about the formof £[J'(t)],whichcan includenonrational functionsof s.In Theorem 12-5, thefunctionisPCSEP,butnosuchstipulationisrequiredforEqs. (12-35)and(12-36). 12-6.Approximationof J(t)byPolynomials.It issometimesincon- venient tofindan exact Laplacetransform of agiven function;or some- times itis impossible, as when the function isspecified graphically or as a tabulatedset ofnumbers.This factlendspertinencetothecommonly usedmethodofapproximating J(t)byapiecewisecontinuousfunction made upof sections ofpolynomials.Weshall now consider theLaplace transformof such afunction.Assume that each section is an nth-degree polynomial.Allderivativesof f(t)willbedefinedinthe sensegivenin Theorem12-6; and each derivative,uptoand includingthenth,can be FURTHERPROPERTIESOFTHELAPLACETRANSFORM371 discontinuousattheendsofthesections(pointsTie)'However,the (n+ l)st derivative willbe zero,because each polynomial is of degreen. Equation(12-35)willbeappliedtothesuccessivederivatives,untilthe (n+ l)st isreached.The followingnotation willbeused: and fo(t)=f: !'(T) dT f ~ ( t )=f: f"(T)dT ., f(t)- fo(t)=L BOku(t- Tk) i-O ., !'(t)- f ~ ( t )=L Blku(t- Tk) k=O .. j rTeand if the integral (F(s)e'l dt JBr existsforallt.Path Br istotherightofsomeabscissarTe.Existence of this integral is a sufficient condition,but not very enlightening.Con- ditionsthatcanbecheckedbyinspectionaremoreuseful.However, thisintegraldoesprovideonesimplesufficientcondition,arrivedat by recallingthat the abovecan bewritten PV f _....F(rT+dw iT> rTo Now,by recalling Theorem 8-6,it followsthat this integral exists if PV f-....W(rT+ jw)1dwrT> rT. exists,and,finally,thelast integral exists if limIwlIF(rT+ jw)1=0rT> rTe 1.,1-..0 (12-47) Theorem10-11providesanotherusefulsufficientcondition,namely, that F(s)shall bearational functionforwhich the degreeofthe denom- inator is greater than the numerator. Nowsupposethat "'(s)isaLaplacetransform.FromTheorem10-8 weknow,if T0,that F(s)=",Cs)e-· T isaLaplacetransform.Finally,by virtueofthe convolutiontheorem, wecansaythattheproductoftwotransformfunctionsoffunctions satisfyingtheconditionsoftheconvolutiontheorem(Theorem11-3)is itselfatransformfunction.Wecollectandstatetheseresultsinthe followingtheorem: Theorem12-9.A function F(s)is the Laplace transform of some func- tion J(t)if anyone ofthe followingconditions issatisfied: 1.If anumber rTeexists such that F(s)is regular forRe 8> rT.and limIwlIF(rT+ jw)1= 0rT> rT. 1.,1- .. 2.If FC,)isarational function of ,for which the degree of the denom- inator isgreater than thenumerator 3.If "'(,) is8Laplacetransform and F(s)is in the form F(s)="'(s)e-· T whereT0 376COMPLEXVARIABLESANDTHELAPLACETRANSFORM 4.If F(s)= G(s)H(s),whereG(s)andH(s)areeachtransformsof functionssatisfying the conditions of Theorem11-3 12-9.RelationshipsbetweenPropertiesoff(t)andF(s).Theorems 12-7 and 12-8 give asmall amount of information about f(t) , at t=0 and t=00;andTheorem10-11hasalreadybeencitedto showthat certain properties of f(t)are related to identifiable properties of F(s) , namely, that if F(s)isar'1tional function with degree of denominator greater than the numerator,f(t)willbeacombinationoftermsliketne al •Depending onhowdetailedonewantstobecome,manyotherpropertiesoff(t) canberelatedtopropertiesofF(s).Inthissectionweshalldealwith oneparticularcase,ofimportanceinthefollowingchapter.Weshall prove the followingtheorem: Theorem12-10.Let F(s)be ofthe form F(s)= G(s)H(s) whereH(s)istheLaplacetransformofh(t),apiecewisecontinuous functionofexponentialorderao,andwhereG(s)isarational function forwhichthedegreeofthenumeratorisnogreaterthanthedegreeof thedenominator.Then: 1..c-l[F(s)] exists and is piecewise continuous and of exponential order. 2 ..e-l[F(s)]existsandiscontinuousfort> 0ifeitherofthefollowing istrue: a.The denominator of G(s)is greater in degree than the numerator. b.h(t)iscontinuous fort> o. PROOF.SincethenumeratorofG(s)isnohigherindegreethanthe denominator,wecan write G(s)= A+ R(s)(12-48) wherethenumeratorofB(s)isoflowerdegreethanthedenominator andAis aconstant.By Theorem10-11,R(s)has an inversetransform bet)which iscontinuous and ofexponential order al.Now F(s)=AH(s)+ B(s)H(s)(12-49) and sinceH(s)isaLaplacetransform,AH(s)obviouslyhas theinverse Ah(t).Also,B(s)andH(s)haveinverseswhichmeettheconditions oftheconvolutiontheorem(Theorem11-3),andthereforeB(s)H(s)is the transform of Jot b(T)h(t- T)dT But h(t)and bet)areboththeexponentialorder,respectively,oforders FURTHERPROPERTIESOFTHELAPLACETRANSFORM377 0:0anda : ~ .Also,eachisbounded forfinitet,and soeachisdominated by an exponential,as follows: Ih(t)1< Mhea,t Ib(t)1< Mbe a • t Theexpressionontherightcanbedominatedby an exponential(there isnolossofgeneralityif weassumethatah~Q:b),andconsequently .c- 1 [B(s)H(s)]isofexponentialorder.Bynowwehaveshownthat eachtermontherightsideofEq.(12-49)hasaninversewhichisof exponentialorder.The sum oftwo functionsofexponential order isof exponential order,and sowehave proved that .c- 1 [F(s)] existsandisofexponentialorder.Thiscompletestheproofofpart 1 of the theorem. Part2isprovedbyfirstobservingthat,eventhoughh(t)mayhave points of discontinuity, .c-1[B(s)H(s)]=Jotb(T)h(t- T)dT isacontinuousfunctionoft.Thus,whenreferringtoEq.(12-49), .c-1[F(s)]iscontinuousif.c- 1 [AH(s)]iscontinuous.Thisleadsdirectly to the two cases listed under(2)in the theorem,namely: a.If thedegreeofthedenominator of G(s)isgreater than thedegree ofthe numerator,Awillbe zero and then .c- 1 [AH(s)]isidentically zero. b.If h(t)iscontinuous,obviously .c-1[AH(s)]=Ah(t) iscontinuous.Thus part 2has beenproved. Anotherusefultheoremcanbeprovedforfunctionswhichcanbe written G(s)H(s): Theorem12-11.If F(s)isoftheform F(s)=G(s)H(s) whereG(s)isarationalfunctionforwhichthe degreeofthenumerator isngreaterthanthedegreeofthedenominator,andif ,£-1[H(s)]has n- 1continuousderivatives,allofwhicharezeroat t= 0,and if the nthderivativeispiecewisecontinuousandofexponentialorder,then .c-1LF(s)]existsand ispiecewisecontinuous. 378COMPLEXVARIABLESANDTHELAPLACETRANSFORM PROOF.Wewrite F(s)= G(s)s"H(s) s" andobservefromTheorem10-9that ,c-l[s"H(s)]existsand ispiecewise continuous.ThefunctionsG(s)/s"ands"H(s),respectively,meetthe conditionsonG(s)andH(s)inTheorem12-10.Therefore,fromthat theorem,weconcludethat .c-1[F(s)]exists and ispiecewisecontinuous. FromTheorem12-11wecanimmediatelyderivethefollowingcorol- lary,by allowing G(s)to bethe single term s".Thus,wehave: Corollary.If .c-1[H(s)]has n- 1 continuous derivatives,all of which are zero at t= 0, and if the nth derivative is piecewise continuous and of exponential order, then ,c-l[s"H(s)]existsand is piecewise continuous. PROBLEMS 11-1.For each of the followingspecialcasesestablish that the Laplacetransform is an entire function,by actually findingthe transform: (a)let)={~ (c)let)_{ ~ e ' " 0< t T,the abovereduces to _(e T / Rc - 1) e- I / RC vc{t)- T/RCRC T< t (14-5) AsTapproaches zero,the quantity in parentheses approaches1,sothat limvc{t)=_1_ e-I/RC 2'..... 0RC (14-6) Thefactthattheresponsefunctionapproachesauniquelimitisnot surprising,sincethe pulsearea fo T Vo{t)dt=1 (14-7) isconstant as the width approaches zero.In the language of Eq.(14-1), the pulse hasunit strength. R Jll l. Vo o:r FIG.14-1.AnRCcircuitexcitedbya pulse of durationT« RC. A1 - - - - - - - - - - ~ oT ~ - - - - - - - - - FIG.14-2.A triangular unit pulse. Nowletthesamecircuitbedrivenbyatriangularpulseofunit strength,asshowninFig.14-2.Againassumingthatvc(O)=0,the transform ofthe responseis 41- 2e- sT / 2 + e ~ s T V.{s)=RCs + 1T 2 s2 =~(-.!._ ~ C+RC)(1_2e-sT/2+ e- sT )(14-8) T2S2sS+ 1/ RC 412COMPLEXVARIABLESANDTHELAPLACETRANSFORM WeareinterestedonlyintheresponsewhenT< t,whichisreadily seen to be v.Ct)=i2 [t+ RC(e- IIRC - 1)] - :2 [(t -+ RC(e-(I-TI2)fRC- 1)] + i2 [(t- T)+ RC(e-(I-T)/RC- 1)] T< t (14-9) It isseenthatthequantityinparenthesesapproachesunityasT -+ 0, and Eq.(14-6)is obtained in the limit. It being recognized that Twill remain finitein any practical situation, it isusefultoestimatehowsmallTmustbefortheactualresponseto differnegligiblyfromEq.(14-6).Consideringtherectangularpulse, weseethat the correction factor inEq.(14-5)is e TIRC - 1_1+ TIRC ++ ... - 1 TIRC- TIRC T =1+ 2RC+ ... (14-10) whichshowsthat,ifT 12RC « 1,theperuniterrorisapproximately T 12RC.Thus, forexample, if T< 0.02RC Eq.(14-6)givestheexactresponse(forT< t)within about1 per cent. For these two examplesthe limits approachedby the actual responses forT< tarethesame,illustratingthatthestrength,ratherthanthe shape of the pulse,is the determining characteristic. JL oT9'-____ FIG.14-3.A network excited by ageneral unit pulse. 14-3.ImpulseResponsefortheGeneralCase.Letusnowconsider thegeneralcaseofalinearlumped-parameterinitiallyrelaxedsystem driven by an arbitrary positive unit pulse, as illustrated in Fig. 14-3.The pulse function,designated by liT(t),is oneof afamilyofpulseswhich are zero outside the variable intervalTand whichsatisfy the conditions and lim(T OT(t)dt= 1 T-+OJo (14-11) IMPULSEFUNCTIONS413 H(8) is the system function,and ~ ( t )is the response. *Since the system haslumpedparameters,itisdescribedbyasetofordinaryintegro- differentialequationsandH(8)isthereforearationalfunctionof8. Wealsotemporarilymaketheassumptionthat H(8)isoflowerdegree in the numerator than in the denominator.Under these conditions(see Theorem10-11)weknowthatH(8)hasacontinuous inversetransform h(t)=.,c-l[H(8)] In transform functions,the system isdescribedby HT(S)=H(s)I1T(s) (14-12) (14-13) where HT(S)and I1T(s)are the respectivetransforms of hT(t)and8 T (t). The product in Eq.(14-13)suggestswriting the convolution formula hT(t)=fOT8 T (T)h(t- T)dTT< t(14-14) TheupperlimitofintegrationisTbecause8 T (t)isidenticallyzerofor T< t.For simplicity,weshallassumethat8 T (t)isnonnegative forall t,and it isknownthat h(t)iscontinuous.One formofthemean-value theorem forintegralst isusedto give ~ ( t )=h(t - X)fOT~ ( T )dT or,in viewofEqs.(14-11), hT(t)=h(t - X) T< t T< t (14-15) where 0~X ~Tforall t.Sinceh(t)iscontinuous.wecan nowwrite limhT(t)=h(t)(14-16) T-+O * H(s)is the system function defined in Sec.13-15. t The mean-valuetheoremusedhereisslightlydifferentfromthat usuallystated in elementarytexts.Therequiredtheoremisreadilyproved,asfollows:Letu(z) be continuous, a~z~b,and let I1(X)be nonnegative for all x in the interval.Being continuous, u(x) will have alower bound mand an upper boundM.The quantities u(x)- m,M- u(x), and I1(X)are all nonnegative.Therefore fab U(X)V(X)dx- mfab I1(X)dx=fablu(x)- mll1(x)dx!?;0 Mfab I1(X)dx- fab U(X)I1(X)dx=fab1M- u(x»)v(X)!?;0 andthereforemfab I1(X)dx~fabU(X)V(X)dx~MfabI1(X)dx In viewof this inequality, fab U(X)I1(X)dx=u(:c ' )fab V(X)dx whereu(x')hasavaluebetweenmandM.Beingcontinuous,u(x)takesonall values betweenmand Min the interval of integration.Therefore,weare sure that a~x'~b. 414COMPLEXVARIABLESANDTHELAPLACETRANSFORM Inwords,westatethisresultbysayingthattheinversetransform of the system function is the limit approached by the response to ashort pulseofunitarea,asthedurationofthepulseapproacheszero.This generalresultisconfirmedbytheexampleinSec.14-2.Thelimit approachedby the responsewas foundto be and its transform is ~e- t / BC RC 1 RCs + 1 which is indeedthe system functionVc(s)/Vo(s). Let us turn again to the practical situation, in which T can never reach zero.The argument on the right of Eq.(14-15)differs from t by at most the amountT.This factgivesthe approximate formula Per unit error< T I~ ( WI = IT~[logh(t)ll(14-17) AppliedtotheexampleofSec.14-2,thisestimateyieldsT I RC,rather than the previously obtained value ofT /2RC.The factor of2 by which thesetwoestimatesdifferisnot significant,sinceweareobtainingonly an order-of-magnitude appraisal.Equation (14-17) is significant because itprovidesanestimatewhichisindependentoftheparametersofa specificsystem.Ingeneral,theresultshowsthatthemaximumrate of change ofh(t)determines how small Tmust be forthe actual response to differfromh(t)by anegligibleamount. In ordertosimplifythederivation,weassumedthatthepulsewasa nonnegativefunction.However,thisrestrictionisnotnecessary,as can be seen by applying the mean-value theorem separately to each inter- valoftoverwhichthefunctiondoesnotchangesign.Ofcourse,the condition that its integral from 0 to Tshall be unity must be retained. Whentheunit-pulse excitationisshort enoughinduration forh(t)to beindistinguishablefromtheactualresponse,suchapulseiscalleda unit impulseandh(t)iscalledthe impulseresponseofthe system.The unit-impulse functionisnot uniquelydefined;the acceptable durationT dependsonthesystemfunction.Inpractice,apulsewhichisshort enoughindurat.iontoactasanimpulseforonesystemmightbetoo extended to act asan impulse foranother system. The main conclusions reached in this section can be summarized by the followingtheorem: Theorem 14-1. *If a linear lumped system, with a transmission function H(s)whichhasazeroatinfinity,isexcitedbyaunitpulse,thelimit •Asimilartheorem can be stated for distributed systems, forwhich H(s) is a trans- cendental functionof the proper formto have an inverse trallilform. IMPULSEFUNCTIONS415 approachedbytheresponse,asthedurationofthepulseapproaches zero,is h(t).. ,c-l[H(a)] 14-4.ImpulsiveResponse.In the previous section we were careful to specify H(a)as arational functionwith azeroat infinity,therebyensur- ing existenceof ,c-l[H(a)].We now consider asystem forwhichH(a)is rational but with an nth-order pole at infinity, in which case ,c-l[H (a)] does notexist.Clearly,thedevelopmentoftheprecedingsectiondoesnot apply.However,if8 r (t)isdefinedsoastohavencontinuousderiva- tives, all of which are zero at t=0,and apiecewise continuous (n + 1)st derivative,the followingsequence oftransforms will exist: Then wecanwrite ,c[8 T (t)]=AT(a) =SATeS) ThefactorH(s)/s.. +1hasaninversetransform,andwehaveshown thatthefactorinbracketshasaninverse.Accordingly,byTheorem 12-10it isknown that ,c-l[HT(S)]existsand can be designatedby hT(t).· Now let H(s)bewrittenasa partial-fraction expansion (see Sec. 5-15), with the polynomial part written explicitly,giving H(s)= G(a)+ Ao+ A 1 s+ ... + Ans"(14-18) In the above, G(s)is the sum of the principal parts at the finite poles and therefore hasan inverse transform get)=,c-l[G(S)](14-19) Now it ispossibleto write HT(S)= G(S)AT(S)+ (Ao+ Ais+ ... + Ansn)AT(s)(14-20) The firstterm on the right is the product of two functions,each of which has an inverse transform.Accordingly, the convolution theorem applies, giving o t< T Tt (14-21) • Theorem12-11couldbeusedto yield this result directly. 416COMPLEXVARIABLESANDTHELAPLACETRANSFORM Furthermore, in view ofthe existenceofnderivatives of 8 T (t),it follows that = Ao8T(t) ,e-l[A 1 s.:l T (S)]=A 1 8 T (t) ,e-l[A"s".:lT(8)]=A .. 6TTl + a,the integral takes on fixedlimits as follows: Thesecondfactoroftheintegrand iszerofort< T+ T 2 ,andtheinte- grationlimitsputTintherangeTl< T < Tl + a.Accordingly,we seethattheintegraliszerofort< T 1+ T 2.Thesamefactorisalso zerofort> T2+ b+ T,and since ThasTl + aas its maximumvalue, theintegraliszerofort> Tl + T2+ a+ b.Thustheintegralin question isa function of t,with the property of being identically zero for t < Tl + TIandfort>Tl + TI + a+ b.It isapulseofduration 430COMPLEXVARIABLESANDTHELAPLACETRANSFORM a+ b.Wehave yet to showthat its integral isunity,making it aunit pulse. Weproceedby showing that the integral from0tocoisunity,recog- nizing that the integral over this range is the same as the integral over the pulsewidth.Sinceaandb arefinite,the following Laplace transforms, =.c[6 a (t)] =.c[6 b (t)] (14-70) exist.FromTheorem11-3fortheconvolutionintegralandTheorem 10-10 forthe transform of an integrated function,wehave .c{lot [loAiia(T- T1)MX- T - T 2 )dT]dX}= (14-71) Thefinal-valuetheoremcanbeusedonthefunctionontheright,to yield the value ofthe integral from0to00.Weget lim=lim .-.0(T.-.0(T(T = 10'"6a(T)dT10'"iib(T)dT=1 (14-72) Thus,it isprovedthat the integralinquestionisaunit pulse,ofwidth a+ b. Earlier,weplacedrestrictionsonthemaximumvaluesofaandb andthen in Eq.(14-68)allowed them to be small enoughto replace Mt) and 6b(t)by 6(t).Assuming that this has been done, consistent notation wouldbeto imply the same small valuesinthe integral just treated,by leavingoffthesubscripts.Thus,whenaandbaresmallenoughfor Eqs.(14-69)to be valid,wethen alsohave Jotii(T- Tl)6(t- T - T 2 )dT=ii(t- Tl- T 2 )(14-73) WhenthevaluesofthefourintegralsinEq.(14-68)arecombined,the resultisinagreementwithEq.(14-66).Thisfactestablishesthat the integration principles developed in Sec.14-10 can be used in the convolu- tion integral to give results consistent .with the definition ofthe symbolic inversetransforms of functionsofthe form G(s)+ Ao + :A 1 8 +... + A,,8" The proof givenapplies only forthe firsttwo terms of the above,but by followingsimilar arguments the generalcase can be confirmed. 14-12.Superposition.TheconvolutionformulaofTheorem11-3, when applied to a linear system, can be interpreted as an expression of the superposition principle.In the notation ofSec.14-7,the responseofan IMPULSEFUNCTIONS431 initiallyrelaxedlinearsystemisgiven,intransformfunctions,byEq. (14-49),whichwerepeat: R(8)= H(8)F(8) If .e- 1 [H(8)Jexists,fromTheorem11-3itisknownthattheresponse ret)is ret)=Jo'h(t- T)f(-,.)dT (14-74) Toanydesireddegreeofapproximation,theaboveintegralcanbe written N ret)=Lh(t- T,)f(T;).iT, i-O (14-75) In Eq. (14-75), the factor h(t- T,)is the response of the system due to a unit impulse occurringat t=T,.Therefore, h(t- T;)f(T,).iT, istheresponseduetoanimpulseofstrengthf(T;).iT;,andthetotal response,givenbythesummationinEq.(14-75),isthesuperposition oftheresponsesofasequenceof impulseswhosestrengthsarepropor- tionaltof(T)atallvaluesofT less thant.Figure14-7illustrateshow f(T)canbethoughtofasconsisting ofasequence ofimpulses. Inviewoftheextensionofthe convolution theorem, as presented in Sec.14-11,itispossibletoextend thepresentconcepttoincludeim- pulse excitations and systems having impulsiveresponsestoadiscontin- uousdrivingfunction.Inother words,it ispermissiblefor F(8)and H(8),or both, to have various-order - ... 7 ' ~ 1-- ...... - ~ FIG.14-7.Approximationof afunction by asequence of pulses.The strength ofeachpulse isequal tothearea of its rectangular representation. impulse functions in their inverses,solong asthe convolution integral is interpreted in the manner described in Sec.14-11. 14-13.Summary.Inthischapterwehaveattemptedtoprovidea rationaldevelopmentofatopicwhichhassometimesbeenconsidered controversial.Thecontroversyhasarisenbecause,aswehaveseen, discontinuousfunctionsdonothaveordinaryderivativesat thepoints ofdiscontinuityandanimpulsefunctiondoesnotapproachafunction in the limit as the width goes to zero.It is not surprising, then, when we formallyget the transform of the derivative of adiscontinuous function, that we obtain a function which does not have an inverse, in the ordinary 432COMPLEXVARIABLESANDTHELAPLACETRANSFORM sense.In spiteoftheseanomalies,it has longbeenknownthat correct answers can be obtained by apurely formal process,by regarding certain functionsof sas Laplace transforms,even though they are not. No attempt to develop atheory to cover these exceptional cases,based on the usual calculus,can succeed.Twobasicreasonscan begivenfor thisstatement:The basicdefinitionofthe derivative cannot be applied at apoint ofdiscontinuity,and an integral isacontinuous functionof a variable limit of integration.Thus,notrue function - T 1 )can exist forwhich Anotherwaytoput it istosaythattheoperationsofintegrationand differentiationarenoncommutative.Inthetreatmentpresentedhere this difficulty has been overcome by defining as always being of finite width. By adopting this policy, it is possible to proceed without violating any mathematicalorphysicalprinciples,althoughsomesymbolicnotation like=1 isused.Thisnotation isnotprecise,to the extent that here the symbol doesnot mean aLaplace transform.The idea ofkeep- ing the pulse width finite is satisfying from aphysical standpoint, because zero-widthpulsesneveroccurinphysicalphenomena.Inpractice,an impulsephenomenonisalwaysonewhoseduration(intimeor space)is small enough so that variation of other function components is negligible throughout the span ofthe pulse. We have attempted to make a distinction between the impulse response of a system and an impuLsive response.Under certain conditions, namely, in asystem whose function H(s)has an inverse,the impulse response isa clearlydefinedfunction,obtainedwhenthewidthofadrivingpulse goestozero.However,whenH(s)doesnotmeetthiscondition,or when the driving function isa higher-order impulse, there will be impulse componentsintheresponse,givingwhatwehavecalledan impulsive response.An impulsive responsecan beobtained even when there is no impulse inthe driving function.An example ofthis occurs when astep function of voltage is applied to a capacitor, current being regarded as the response.Intheanalysispresenthere,theimpulsecomponentsofan impulsive responseare alwaysoffinitewidth. Itmaybedisturbing' toyouthatweusetheuniquesymbol to imply apulsewhosewidth and shape are left unspecified.Ofcourse,as hasbeenpointedout,theareaunderthepulseistheuniquefeaturein determiningtheimpulseresponse;andwhenaresponseincludesan impulseterm,weunderstandthistobeaconditioninwhichthepulse is so short in duration that its detailed shape isnot observable and isof IMPULSEFUNCTIONS433 noimportance.If weareinterested in pulsedetails,thepulsecannotbe calledan impulse. MentIonshouldcertainlybemadeoftheworkofL.Schwartzand others * inwhichdistributionsareusedinsteadoffunctions.I tisthen possibletodefineanoperation whichisanalogoustodifferentiationbut whichhasmeaningforsituationswhichbearasimilaritytothediscon- tinuous functions.However, in order to apply this theory, it is necessary tochangetheconceptualmodelsusedtorepresentphysicaldevices,in suchawaythattheircharacteristicsaredescribedbydistributions ratherthanfunctions.Therefore,theredoesnotseemtobeanyway toappendthetheoryofdistributionstothepresenttheory,whichis built onthe concept ofafunction. PROBLEMS 14-1.In Fig.14-1let the excitation beaunit voltage pulse -sm- { I.rt 110(t)=~T 0< t< T T< t Obtain theresponsetothispulse,andthelimitofthisresponse,asTgoestozero. Compare your result with the limit obtained in the text. 14-2.Showthatapulsecanbenegativeduringpartofitsduration,andstillbe considered aunit pulse,so long as its area isunity. 14-3.For eachofthe circuits of Fig.P14-3 what duration of excitation pulse 11,(t) would be adequately short,sothat the pulse couldbe considered an impulse? + (a) (b) FIG.P14-3 14-4.Consider asyst"m whoseresponse functionis • L. Schwartz, "Theorie des distributions," vols. I, II, Hermann &Cie, Paris, 1957; I.Halperin,"IntroductiontotheTheoryofDistributions,"UniversityofToronto Press,Toronto,1952;M.J.Lighthill,"AnIntroductiontoFourierAnalysisand GeneralizedFunctions,"CambridgeUniversityPress,NewYork,1958;SirGeorge Temple,J. LondonMath.Soc.,28:134-138(1953). 434COMPLEXVARIABLESANDTHELAPLACETRANSFORM Designaunitexcitationpulseoffinitedurationwhichcanbeusedinanexact application of the Laplace transform theory,but such that the response differs negli- gibly from the impulse response.If the system function were 1 H(s)=-- I+s how could the pulse specification be changed? 14-6.Show that the driving-point response of a passive network cannot have higher than asecond-order impulse function(doublet)in its impulse response. 14-6.Provethatallordersofsingularityfunctions,higherthanthefirst,have zeroarea. 14-7.What is the response,to asecond-order impulse of unit strength, of a system whose functionis ()H( )=(8+ 0.5)(8+ 1.5) a/I(8+ 1)(8+ 2) 8+0.5 (b)H(8)=(8+ 1)(8+ 2) 14-8.Discussthedifferencebetweentheconceptsofanimpulseresponseand an impulse function. 14-9.Refertothefunctionsdefinedin(a)and(b)inProb.12-12.In eachcase specifydf(t)/dt,usingsymbolicimpulsefunctions.Also,obtain.c[df(t)/dt],and obtaintheinverseofthistransformdividedby8.It shouldbethesameasthe original function. 14-10.For the followingexamples,obtain the responsetoaunit impulseand to a unit step, and observe that the formeristhe derivativeof the latter: 1 (a)H(/I)=(s+ 1)(8+ 2) 8+1 (0)H(8)=8+ 2 /I+ 1 (b)H(s)=8' + /I+ 1 /I- 1 (d)H(s)=8+ 2 CHAPTER15 PERIODICFUNCTIONS 16-1.Introduction.In many importantphysical systems thed r i v i n ~ functionisperiodic.Intheclassicaldevelopmentofthetheoryofthe behavior of such systems,the sinusoidal excitation occurredfirst,and in manyrespects,itisthesimplest.It wasrecognizedearlythatthe solutioncouldbebrokenintotwoparts,usuallycalledthetransient responseandthe steady-state response,aswehavepointedout insome detailinChap.1.Inthatchapterwealsoshowedthatforanon- sinusoidal periodic driving function the Fourier series provided a solution. FIG.15-1.Example of a.periodic function. TheFourier-seriessolutionforthenonsinusoidalcaseiscertainlyof greattheoreticalimportance,providinganimportant linkinthetheory oflinearsystems.However,inmanycasestheFourierseriesisnot convenientforcomputation,owingtoslownessofconvergence.Inthe presentchapterweshallbeconcernedwithamethodofobtainingthe responseofasystemtoaperiodicdrivingfunction,whichispreferable when specificnumerical answersare required. Ourattentionwillbeconfinedtothosecaseswherethedrivingand responsefunctionsareatleastpiecewisecontinuous.Thegistofthe methodisverysimpleandcanbedescribedinterms ofthe triangular functionshowninFig.15-1.Thisfunctioncanbedescribedbythe Fourier series I(t)= !(Sin ",t- ! sin3",t + .!.. sin 5",t...) 11"2925 (15-1) andthisseriescouldbeusedtocalculatevaluesofthefunction,but withsomeeffort if ahighdegreeofaccuracy isrequired.Computation 435 436COMPLEXV ARIABL}]SANDTHELAPLACETRANSFORM of such aseries ismore difficult if it representsadiscontinuous function, becauseconvergenceisthenslower.Nowweobservethesimpleidea that this same functioncanbespecifiedby the pair offormulas 2'If' - wt 'If' f(t) 2 t fTc(16-7) n=O THEZTRANSFORM447 where(1'.istheabscissaofconvergence.WestipUlatedthat J(t)should beof exponential order.Therefore,corresponding to an arbitrary small number E> 0,there is anumber Nsuch that, if a> ao, If(nT)le-a .. T< E and If(nT)le- w .. T =If(nT)le-a"Te-(w-a)"T< Ee-(w-..)"T (16-8) whenn> N The series isknown toconvergeif (1'> a.Therefore,the seriesinEq.(16-7)con- vergesabsolutely,andthereforealsoconverges,for(1'>ao.It canbe showntodivergefor(1'< ao.Thus,theabscissaofconvergenceofthe Laplace integral,which is also the abscissa of convergence of the series, is (1'.= ao andcanbedeterminedfromthebehavioroff(t)at infinity,sinceitis thisbehavior that determinesao. It isnowconvenientto makethevariablechangez= e· T ,asdefined inEq.(16-1),whichtransformstheaxisofconvergenceRes=(1'.into the circle z=eW,Tei"r Izl=ea,T (16-9) In the z planewenow have anew function F.(z)=F*(s) and,fromEq.(16-7), .. F.(z)=L f(nT)or (16-10)* .. -0 Bytheprincipleofanalyticcontinuation,it isknownthat F.(z)isan analytic function; and the series on the right of Eq.(16-10)is the Laurent expansionabout theorigin.F.(z)iscalledtheZtransformof J(t).A shorthandnotationissometimes useful,corresponding to F(s)=,c[f(t)]. Accordingly,wedefine F.(z)=Z[f(t)](16-11) to mean the functiondefinedbyEq.(16-10). •In this equation it isunderstoodthat 1(0)means 1(0 +). 448COMPLEXVARIABLESANDTHELAPLACETRANSFORM 16-3.Z Transform.of Powers of t.The Ztransform oft",as aseries, is foundfromEq.(16-10)to be .. Z(t")= 2: n"T"z-aIzl> 1 .. -0 .. =T1.2: nk-1Tk-lnz-(n+l) (16-12) n=O .. and also Z(tk-l)=2: nk-1Tk-l1.-a Izl>1 (16-13) .. =0 Observethat the seriescan bedifferentiatedterm by term,and so 11.1>1 Comparing with Eq.(16-12),weget the recurrenceformula Z(t")=- T1.11.Z(t k - 1 ) For k=0,wehave 00 Z(I)=2: 1.-ft 11.1>1 .. =0 which isrecognizedas theLaurent expansion of Z 1.- 1 (16-14) Thus, asequence of formulasisobtained fromEq.(16-14),as follows: 1. Z(I)=-- 1.- 1 T1. Z(t)=(z_1)2 Z(t2)=T2 1. (1.+ 1) (1.- 1)3 (16-15) SincetheZtransformsareobtainedfromtheLaplacetransforms merelybyachangeofvariable,weconcludethattheZtransformofa sum isthe sum ofthe transforms.Also,theZtransform ofaconstant times a function of t is equal to that constant times the Ztransform of the THEZTRANSFORM 449 function.Accordingly,it isseenthatthe Ztransformofapolynomial in tis arational functionofz. 16-4.ZTransform of a Function Multiplied by e-G'.From Eq. (16-10), wehave Thus, if .. Z[e-f(t)]=L f(nT)e-G"Tz-" ,,-0 .. =L f(nT)«f'Tz)-" ,,-0 F.(z)=Z[f(t)] it followsfromEq.(16-16)that Z[e-a'f(t)]=F.«f'TZ) (l6-16) (16-17) ReferringtoEqs.(16-15)andtaking f(t)=tk,wehavetheadditional formulas (16-18) Thesearerational functionsofz,andasum ofafinitenumber ofthem willberational.From the regionofconvergenceofEq.(16-10)wealso knowthatthepolesofthisfunctionwilllieinsideacircleofradius e aoT ,where f(t)isofexponential order ao. Sincefunctionsofthistypeareusedsofrequentlyinanalysis,itis usefultohavethefollowingtheorem,forwhichtheabovedevelopment constitutes aproof: Theorem16-1.If f(t)is asum of afinitenumber of terms ofthe form tkandtke-G',the sumbeing ofexponential order ao,the Ztransform isa rational functionof z,havingpoles insideacircleof radius e aoT • AusefulcorollaryisobtainedbyreferringtoTheorem10-11,which tells us that, if F(s)is arational functionwith azero at infinity,then its inverse will be a function of t such as is described in Theorem 16-1.Thus, wehave the followingcorollary: Corollary.If F(s)isarational functionwith azeroat infinity,and if aoisthelargestrealpartofthepolesofF(s),thenZ{.c- 1 [F(s)Jlisa rational functionofz,with poles lying insideacircleofradiuse«.T. 450COMPLEXVARIA-BLESANDTHELAPLACETRANSFORM 16-6.The Shifting Theorem.Consider a function f(t)for which the Z transform F.(z)isgiven by F.(z)::;:f(O)+ f(T)z-l+ f(2T)r t +... + f(NT)r N +... Izl>e",T(16-19) Suppose that wenow findthe Ztransform of the shifted function f(t- NT)u(t- NT) wherethe shift isan integral number ofsamplingperiodsNT.In view ofthe unit step u(t- NT),this functioniszeroforte",T The series converges uniformly, for Izl~R'>e",T,and therefore the limit ofthe seriesasIz\~00can beobtainedby taking thelimit ofthe indi- vidual terIDS,giving limF.Cz)=f(O)(16-21) 1-1-" For considerationofthe final-valuetheorem,weareinterested inthe behaviorofF.Cz)asz -+ 1,aswemightsuspectbyrecallingthefinal- value theorem ofthe Laplacetransform,which involves the point 8= o. If f(t)isofexponential orderao,whereao< 0,the Z-transform series .. Z[f(t)]=L f(nT)z-N n-O (16-22) converges for\z\>ea,T.Sinceao e",T,asshowninFig. 16-2,and convert the above integral to acontour integral forwhich Rei'=z 1 d()=-;- dz JZ giving f(nT)=2 ~(F.(z)zn-l dz(16-28) 1I"J}e. The above integral is the inversion FIG.16-2.The integration contour usedformulafortheZtransform.We fortheZ-transforminversionintegral. observeparticularlythatitgives f(nT)onlyforintegral values of nj it doesnot uniquely specify f(t).If F.(z)issingle-valued,thecalculusofresiduesisreadilyavailableto evaluate this integral.Coenclosesall the singular points of F . ( z ) ~ It isinteresting to observeadifferencebetween theLaplace inversion .c-1[F*(s)]=f(t)p(t)= ret) THEZTRANSFORM 453 andtheresultjustobtained,wherebyl(nT)isobtainedfromF.(z). Werecognizef*(t)asasymbolicfunctionwhichhasanimpulseof strengthl(nT)att= nT.Equation(16-28)givesthemoreuseful l(nT) ,whichwillberegardedastheinverseofF.(z),andaccordingly wewrite l(nT)= Z-I[F.(Z)] Although there isan exact functionalequivalence H*(s)= F.(z) theinverses£-I[H*(s)]andz-I[F.(z)]aredifferent. Asan illustration ofthe inversion formula,consider Tz F.(z)=(z_1)2 (16-29) which is known to be the Ztransform of I(t)=t.The inversion integral gives - 1fTz.. -1- ~fTz" l(nT)- -2·(1)2zdz- 2·(1)2 dz 7rJCOZ- 7rJCOZ- with the understanding that Cohas a radius greater than 1.The second- orderpoleat1isthereforeenclosed,andtheintegralisequalto27rj times the residue,where Residue=.!:..Tz" I=nTz ..-ll=nT dz0-11.-1 Thus,the expected result l(nT)=nT isobtained. Weconcludetheseremarksabouttheinversionintegralwiththe observationthattheinversionintegralisnotnormallyneeded.·For agiven F.(z)it is necessary only to expand in aLaurent series about the origin,saybyadivisionalgorithm.Thesuccessivevaluesofl(nT) are the coefficients in this series. 16-8.PeriodicPropertiesof F*(s),andRelationshiptoF(s).Inthis sectionwereturntoaconsiderationofF*(s),recallingthat it isrelated to the Ztransformby F*(s)=F.(z) By virtue ofthe Laurent-series representation for F.(z),weare assured that F.(z)isan analyticfunctionof z.Since e-T=e(o+jhn/T)T 454COMPLEXVARIABLESANDTHELAPLACETRANSFORM it followsthat theZtransform isaperiodicfunctionofs,whichfactis put into evidenceby writing F*(s)= F* (s + j2;7r) (16-30) CorrespondingtoeachsingularpointofF.(z),therewillbeaninfinite set of singularities of F*(s).Specifically, if Zl= Xl+ jYl isa singularity of F.(z),F*(s)issingular at s"= ;logZl= log(Xl'+ Yl')+ j; (tan- l + 27rn) wherentakesonallintegralvalues.If F(s)isrational,implyingthat F.(z)isalsorational,weseethat F*(s)ismeromorphic. Inviewoftheabovecomments,whenF(s)isrational,wesuspect that F*(s)canbe expressedinaMittag-Leffler(partial-fraction)expan- sion,in the manner described in Sec.5-16.The objective of this section is to show that this is true and to obtain the expansion.F(s)is assumed to be agiven rational function,with a zeroat infinity.Assume that it is written in partial-fraction form,for which wewrite three specific terms as follows: F(s)=. +_G_+ s-a + (s(:J)'+ . . . +c+... (s- 'Y)I (16-31) The totalnumberofterIDSisfinite.Thecorresponding f(t)functionis f(t)=...+ ae'"+ . . . + + . . . + t 2 e"r'+ . ..(16-32) and the Ztransform,according toEqs.(16-18),is az F(z)=... ++ ... •z- e"T Sincethe factorse6'l'and e"r'l'causeonly ascale changein the z variable, Eq.(16-14)isapplicable, giving bTe6'l'z (z- e6'l')1 cT1e"r'l'z(z+ e"r'l') 2(z- e"rT)1 ) dzz-e(J'l' dcTe"r'l'z - Tz - .-=-,----=7- dz[2(z- e"r'l')I] (16-34) THEZTRANSFORM455 withsimilarrelationsforhigher-orderterms.NotingfromEq.(16-1) that ds1 dz=Tz weseethatinthevariablestheabovebecomessomewhatsimpler,as follows: bTePTe,Td(be,T) (e,T- ePT)2=- dse,T- e PT cT2e.,Te3T(e,T+ e.,T)= .!.~(ce,T) 2(e'T- e.,T)32 ds 2 e,T- e.,T (16-35) Thus,by usingEqs.(16-33)and(16-35),aformula forF*(s)is ae,Td(be,T)1d 2 (ce,T) F*(s)=...e,T_eClT+. . .- dse,T- ellT+"2 ds 2 e'T- e.,T + . ..(16-36) This formis usefulbecause it showsthe importance ofthe firstterm, the other terms being derivatives of a similar function of s.Accordingly, the function ae,Ta e· T + e ClT 1- e(.CI)T (16-37) willbe studied indetail.By theprincipleestablished in Sec.5-16,this canbewrittenasaninfinitesumoftheprincipalpartsateachofthe poles,which occur at '2 s"=a+J ::" J. Eachpoleissimple,with residue ra(s- a- j2rn/T)a 1m1_e(,CI)T=-T ............ Thus,theMittag-LefflerexpansionofEq.(16-37)is .. ae,Ta~1 e'T- e ClT = T'-'s- a- j2rn/T (16-38) n--oo Thisexpansioncanbedifferentiatedtermbyterm,andso,in viewof Eqs.(16-35),fromthe above weget the additional expansions and • b T e ~ T e ' Tb"1 (e· T - ~ T ) '=TL.,(a- a- j2rnpt). tI--· . cTle.,Te'T(e,T+ e"")c~1 2(e,T- e.,T)I=T'-'(8- a- j2rn/T)' n--- (16-394) (16-39b) 456COMPLEXVARIABLESANDTHELAPLACETRANSFORM withsimilarresultsforhigher-orderterms.Equation(16-36)cannow be written .1[ F*(s)=T. 1+ s- ex- j21rn/T ... +b '\'1+ '-'(s- ex- j2rn/T)2 ... + c2:(8- ex_lj2rn/T)3+... ] n--oo .. =- ... ++... 1[2:a Ts- ex- j21rn/T n ...-oo + (s_ex_b j21rn/T)2+ ... + (s_ex_c j2rn/T)3+ ...] ReferencetoEq.(16-31)showsthatthiscanbegivenbythesimpler expression (16-40) n__ oo Thisresultcanbeobtained formallyin amuch simplerwaybyusing s-planeconvolutionofthetransformsoff(t)andpet).However,the resultisanintegralwhichisnotknowntoconverge,andsotheabove more extended derivation ispresented.The followingtheorem has been proved: Theorem16-5.If F(s)isarationalfunction,withazeroat infinity, the Ztransform,as afunctionofs,isgivenby ... F*(s)=~2:F (s - ~ 2 1 r n ) n_-oo 16-9.Transmission of aSystem with SynchronizedSampling of Input andOutput.AsapracticalapplicationoftheZtransform,considera system function H(s)which is rational and has azero at infinity.If f{t) is sampled by multiplying by pet), the transform of the output is R(s)=H(s)F*{s)(16-41) The output ,£-l[R(s)] is continuous.Now suppose that the output is also sampled,usingthesamesamplingfunctionpet).ItstransformR*{s) THEZTRANSFORM457 is to be found.From Theorem16-5,Eq.(16-41)becomes and so ee R*(s)= 2:H(s- 2:F[ s-+ k)](16-42) k--- In the second summation, forfixedk,wecan write v= n+ kand sum u from- 0()too(),with nochangeinthe sum.The result is orR*(s)=H*(s)F*(s) (16-43) This resultcan alsobe expressed in terms of Ztransforms: R.(z)=H.(z)F.(z)(16-44) The function H*(s) is called the sampled transfer function of the system. Uponrecalling that h(t)=£-1[H(s») isthe impulse response,it is evident that h(nT)=z-l[H.(z»)(16-45) In viewofthecommentsattheendofSec.16-7,it isobservedthat Eqs.(16-43)and (16-44)are not exactly equivalent.Inversion ofR*(s) by theLaplace inversion integral gives r*(t)=£-1[R*(s») whereasthe ZinversionofR.(z)gives renT)=z-1[R.(z») whichare relatedby e r*(t)=L r(nT)5(t- nT) n-O Neither inversion definesret)forvaluesoft other than integral multiples ofT. 16-10.Convolution.TheZtransformprovidesausefulconvolution formula.In the notation ofSec.16-9,wewrite R.(z)=H.(z)F.(z) 458COMPLEXVARIABLESANDTHELAPLACETRANSFORM and directourattentiontowardrelatingZ-l[R.(z)]to the inversetrans- formsofH.(z)and F.(z).Equation(16-10)gives .. F.(z)=I f(uT)z-u .. =0 .. H.(z)=I h(vT)z-· .-0 and fortheir product wehave .... H.(z)F.(z)=I h(vT)z-·I f(uT)Z-U .=0.. -0 .. =IA,.z-" n=O (16-46) whereA ..isthesumofallproductsh(vt)f(ut)forwhichv + u=n. This sum isgivenby either ofthe following: .. A ..=I h(kT)f[(n'- k)T] k=O (16-47a) n An=Ih[(n- k)T]f(kT) k-O (16-47b) Comparing Eq.(16-46)with Eq.(16-10)shows thatAn is Z-l[H.(z)F.(z)] att=nT.ButrenT)isourdesignationfortheaboveinverse,and therefore wecan write the followingconvolution theorem: Theorem16-6.If f(t)and h(t)have respectiveZtransforms F.(z)and H.(z),thenthe inverseZtransformoftheproduct F.(z)H.(z)isafunc- tion renT)whichcanbe expressedas .... rent)=- I h(kT)j[(n - k)T]orrent)=I h[(n - k)T]f(kT)(16-48) i-Oi-O 16-11.TheTwo-sidedZTransform.TheZtransform can be extended to those situations for which the Fourier transform and two-sided Laplace transforlDSare used,by definingatwo-sidedZtransform . F ~ ( z )=If(nT)z-" (16-49) n.--. THEZTRANSFORM459 The abovecanbewritten asthe sumoftwo ordinary Ztransforms: F,.(z)= F.l(Z)+ F02 G)(16-50) .. F I1 (z)=L f(nT)z-n n-O where .. (16-51) F02(Z)=L f( -nT)z-n n-l The seriesforFa2(1jz)convergesinsideacircle,whereasthefirstseries converges outside acircle.If 0"1< 0"2,there isaringofconvergence. In regard to the inversion integral and uniqueness, the situation is very similartothetwo-sidedLaplacetransform.Theinversionformulais simpler to derive than for the single-sided case.Referring to Eq.(16-49), if z=Re i6 .. wehaveF,.(Re i6 )=lf(nT)R- n e- in6 (16-52) n- -00 which shows thatf(nT)R-n is the general Fourier coefficient of the expan- sionofF 20 (Re i ').Thus,fromtheformulaforFouriercoefficients,we immediately get f(nT)=2 ~10 2 "F20(Rei6) R n ein6dB whichconverts to the contour integral f(nT)=21.(F 2 .(z)zn-l dz 1rJJc. (16-53) Thisformulaisidenticalin formwithEq.(16-28).However,wenote thatnowtheradiusofCoisrestrictedtobeingintheannularregion between the two circles of convergence.If a radius is used which carries CooOutsidethisregion,theinversionformulawillgiveadifferentf(t), but onewhichhasthe same two-sided Ztransform.In other words,in similaritywith the two-sidedLaplacetransform,the two-sidedZtrans- formisnotuniquelyrelatedtothe f(t)function,unlessthecontourof integrationoftheinversionintegralisspecified.Thecorresponding situation fortheLaplace casei ~discussedin Sees.10-17and 10-18. 16-12.Systems with Sampled Input and Continuous Output.If a sys- temhasasampledinputbutanonsampled(continuous)output,the LaplacetransformoftheoutputisgivenbyEq.(16-41)ratherthan Eq.(16-43).Ztransforms are particularly suited to Eq.(16-43)because H*(s)= H.(z)isrationalinzif H(s)isrationalins.However,H(s) 460COMPLEXVARIABLESANDTHELAPLACETRANSFORM is not rational in z,and forthat reasonEq.(16-41)isnotsimplifiedby introduction ofthe Ztransform.It is best to remain with the s variable, Laplace theory being used to obtain .c- 1 [R(s»).Of oourse, F*(s) is trans- cendentalin s,but thisislessdisturbing than the multivaluednessthat occurs if wetry to use 1 s= "flog II intherationalfunctionH(s).Furtherthoughtonthisquestionwill showthat,if weattempttogetaninversionformulalikeEq.(16-28) forcontinuous functions,the integrand ismultivalued except at t= nT. It ispossible,however,toadapttheZ-transformtheorytoobtain informationaboutacontinuousoutput,byperiodicallysamplingthe output at values of t=T,T+ T, T+ 2T, etc., where the variable param- eter Tisin the range 0~T~T.In thisway,the output can bedeter- mined at any sequence ofpoints,arbitrarily located between the original samplingpoints. The Laplacetransform ofthe continuous output is R(s)=H(s)F*(s)(16-54) whereweareassumingthat theinput issampledat 0,T,2T,etc.We recallEq.(16-10)and write OD H(s)F*(s)=L H(s)f(kT)e-· kT .1:-0 Since h(t)=.c- 1 [H(s)],wehave,according to Theorem10-8, .c- 1 [H(s)e-· k TJ=h(t- kT)u(t- kT) (16-55) (16-56) Thus, assuming that we can take inverse transforIns inside the summation ofEq.(16-55),the output ret)=.c- 1 [R(s)]is OD ret)=Lf(kT)h(t- kT)u(t- kT) k-O (16-57) Although this isthe functionwewant,it isnot in aconvenient form for computation.It isnothingmorethan the summationofresponsesdue to impulsesofstrength f(kT)occurringat t=kT,aresultwhichcould have been derivedwithout the aid ofZtransforIns. Nowletussampleret)atthepointsT,T+ T,etc.,bymUltiplying ret)byasamplingfunctionlikepet),butforwhichtheimpulsesare shifted an amount T.Thus,in similarity with J!;q.(16-4),wedefine .. p(t,T)=l6(t- T- nT) .. -0 (16-58) THEZTRANSFORM461 whichyieldsthe sampled output r*(t,r)=r(t)p(t,r)(16-59) In comparisonwith Eq.(16-7)the Laplacetransform of this is .. .e[r*{t,r)]=Lr{r + nT)e-·(r+lIT) 11-0 (16-60) The function .e[r*{t,r)]isreally superfluous,aswenow show by defining .. R.{z,,,)=l r(1'+ nT)z- .. -0 (16-61) where"= TIT.R.(z,,,)is called the modified Ztran8form of ret).Thus, (16-62) The factore-" showsthat R.(z,,,)istheLaplacetransform of the output functionshiftedanamount l'totheleftandthensampledat t = 0,T, 2T,etc.Equation(16-61)yieldsthesameconclusionby emphasizing that R.(z,1/)is the ordinary Ztransform ofret+ 1'). Equation(16-57)canbeusedin Eq.(16-61).Thiswouldintroduce aunit-stepfactoru[r + (n- k)T)whichiszerofork> n.Therefore, theuppersummationlimitinEq.(16-57)canbechanged to n,giving ..n R.(z,,,)=L l f{kT)h[r + (n- k)T]z-1I n=Ok-O (16-63) ThisresultissimilarinformtoEq.(16-46)whenEq.(16-47b)isused forA".This factcanbeusedto establish that Eq.(16-63)canbeput in aformsimilar tothe firstformofEq.(16-46),namely, .... R.(z,1/)=L h(r + vT)r"lf(uT)ru .,-0u-O (16-64) ThesecondsummationontherightisF.(z).Thefirstsummationon the right is similar to Eq.(16-61),and sowedefine the followingmodified Ztransform ofh(t): .. H.{z,1/)=L h(r + vT)z- v-O (16-65) The finalresult can thusbewritten R.(z,1/)=H.(z''1)F.(z)(16-66) The function H.(z,1/)is readily found fromEq.(16-65),being the normal Ztransform ofh(t + 1'). 462COMPLEXVARIABLESANDTHELAPLACETRANSFORM In concept,thisresultisverysimple.Toobtainasolution,wepro- ceed as with ordinary Ztransforms, but using an ordinary Ztransform of h(t + 1'),namely, H.(z,7J)=Z[h(t + 7JT)l,in place of H.(z).The inverse oftheresultingZ-transformfunctionthengivesvaluesofr(7JT+ nT) at samplingpointsdisplacedanamount7JT= l' fromtheoriginalsam- pling points. It isinterestingtoobservethatEq.(16-44)canbeobtainedbythe aboveproof.This isevidentwhenwenotethat,if7Jgoestozero,the modifiedtransform reducestoan ordinaryZtransformandEq.(16-44) becomesaspecialcaseofEq.(16-66).In fact,to proveEq.(16-44)by thepresentmethod,althoughperhapsmorecomplicated,ispreferable becauseit doesnot require F.(z)and H.(z)to berational functions. 16-13.DiscontinuousFunctions.Throughoutthediscussionofthe Laplacetransform,muchattentionisgiventothepossibilityofJ(t) havingpointsofdiscontinuity.Incontrast,thediscussionoftheZ transform has been based on the assumption thatJ(t) is continuous.This is not a significant omission,because f*(t)isinsensitivetodiscontinuities occurringbetweenthesamplingpoints.Furthermore,if J(t)shouldbe discontinuousatasamplinginstant,thetheorypresentedisstillappli- cable.In that case, if there is a discontinuity at niT, it is necessary only to replace J(7J i T)by J(niT+),whereverit occurs. 16-1.Obtain the Ztransforms of (a)sin bt (c)cosbt PROBLEMS (b)eG'sin bt (d)cosatsin bt and locate their singular points in the IIplane. 16-2.Show that (nT+ r'[F.(z)]=)0/*(t) dt where nT + signifies avalue slightly greater than nT.Both sides of this equation are functionsof n. 16-S.Let the input functionof asystem be f(t)=e- G ' and lIupposethat the system functionis .1 H(s)- 8+ b The system is initially relaxed. (a)Assuming sampling intervals of duration T, and using series expressions for the appropriate Ztransforms, find the first three terms of the synchronously sampled out- put function(corresponding to t=0,T,2T). (b)Usingtheinversionintegral,obtainageneralexpressionfortheoutput func- tion requiredin(a),and compare with the threevalues obtained in(a). THEZTRANSFORM 16-4.Let a system be described by afunction and assumeadriving function , H(,)- ,+ b I(t)- sin bt 463 sampledat integralmultiplesofT.Usingtheinversionintegral,obtainanexpres- sionforthesynchronouslysampledresponse,assumingthatthesystemisinitially relaxed. 16-6.Obtain the solutiontothe systemspecifiedinProb.16-3,by usingthe con- volution theoremforZtransforms. 16-6.Showthat at time nT< t< (n+ I)T thenonsampledresponseofthe sys- tem definedin Prob.16-3 is 16-7.Obtain the solutionforthe nonsampledresponse,forthe situation described in Prob.16-4. 16-8.A system wlth input function J(t)whichhas been sampled at 0,T,2T,.. isto operate on this sampledinput in such away as to give an output of the form { J(O) ret)=J(T) 1(2T) 0< t< T T< t< 2T 2T< t< 3T Showthat a system function _ 1- e- 02 ' H(s), will meet these requirements. 16-9.A system is describedby the function 1 H(,)=-- ,+1 AssumethattheinputI(t)issampledatintegralmultiplesofT.Theoutputis unsampled.If r(nT)is the corresponding sampled output, show that ret)=e-(l-.. TlJ(nT) where nT < t< (n+ l)T. 16-10.For the general function t>O tO ae;O 2Y; (v;-=li - v'B=(i) ae;O . ra;60/4, (err._1) + "481'14,, r'(I)- log, , r(n) log, ," n>0 a"Va>+ s' -v'8+2li - vB "lis+ 2a+ Va n>-1 a> 0,n> -1 ae;O n> 0 APPENDIX TABLEOFFUNCTIONSANDTHEIRLAPLACETRANSFORMS(Continued) f(t) 1 Vi (t+ a) 1 vt +a 1 t+a Si(at) erfy(ji erf(at) 1- e r f ~ Vi Ei (-at) Ci(at) F(8) ~e O '(1- erfv'ii8) Va ~e··(1- erfvas) a>O a>O -eO. Ei(-as)a>O !-I! e"14.(1_erf_8_) 2'Ia2Va ! tan-II! 88 a>O ! e,· 14 .' (1- erf .!...) 82a 18+ a - Blog-a- a> 0 1at+ 8' - 2s log a.- 2v'8 (8+ a) .y'ii3; 2W(8 +a) a>O a>O a>O •Brackets denote the greatest integer. a>O 467 APPENDIXB 1d" p.(x)=2"n! dx"(x'- 1)· Legendre polynomial Laguerre polynomial J,,(x)=(-;?" Jo" e fu .", cosnu duBessel function erf x=- e-u'du 2h'" ...;;0 .. Eix=- -du Iu J I. S ·SInxu d IX=-- U ou • CIX=- --du Iu ModifiedBeBBelfunction Error function Exponential integral function Sineintegral function Cosineintegral function 468 BmLIOGRAPHY Worksdealingmainly with functi0n8of acomplexvariable: Churchill,R.V.:"IntroductiontoComplexVariablesandApplications,"2ded., McGraw-HillBookCompany,Inc.,NewYork,1960. Copson,E.T.:"TheoryofFunctionsofaComplexVariable,"OxfordUniversity Press,NewYork,1935. Franklin, Philip: "A Treatise on Advanced Calculus," John Wiley&Sons,Inc.,New York,1940. ---: "FunctionsofComplexVariables,"Prentice-Hall,Inc.,EnglewoodCliffs, N.J.,1958. Guillemin,E. A.: "Mathematics ofCircuit Analysis," John Wiley&Sons,Inc.,New York,1949. Kaplan,W.:"Lectures on Functions of aComplex Variable," University of Michigan Press,AnnArbor,Mich.,1955. Knopp,K.:"Theory of Functions,"DoverPublications,NewYork,1945-1947. Macrobert, T.M.: "Functions of aComplex Variable," St.Martin's Press,Inc.,New York,1950. Nehari,Z.:"ConformalMapping,"McGraw-HillBookCompany,Inc.,NewYork, 1952(Dover Publications,Inc.,New York,1975). Worksdealingmainly with theLaplace and related transforms: Aseltine,J.A.:"TransformMethod in Linear System Analysis,"McGraw-Hill Book Company,Inc.,NewYork,1958. Bochner,S.:"VorlesungenfiberFourierscheIntegrale,"ChelseaPublishingCom- pany,NewYork,1948. ---and K.Chandrasekharan: "Fourier Transforms," Princeton University Press, Princeton,N.J.,1949. Campbell,G.A.,andR.M.Foster:"Fourier IntegralsforPracticalApplications," D.VanNostrandCompany,Inc.,Princeton,N.J.,1948. Carslaw,H.8.:"Theory of Fourier Series and Integrals," 2d ed., St.Martin's Press, Inc.,NewYork,1921(3d ed.,DoverPublications,Inc.,NewYork,1950). --- andJ.C.Jaeger:"OperationalMethodsinAppliedMathematics,"Oxford University Press,New York,1941. Cheng,D.K.:"Analysis of Linear Systems," Addison-WesleyPublishingCompany, Reading,Mass.,1959. Churchill,R.V.:"FourierSeriesandBoundaryValueProblems,"McGraw-Hill BookCompany,Inc.,NewYork,1941. ---: "OperationalMathematics,"2ded.,McGraw-HillBookCompany,Inc., NewYork,1958. Doetsch,G.:"TheorieundAnwendungderLaplaceTransformation,"Springer- Verlag,Berlin,1937. 469 470COMPLEXVARIABLESANDTHELAPLACETRANSFORM ---: "Handbuch der LaplaceTransformation," vols.I-III, Birkhauser, Stuttgart, 195(}-1956. ---: "Einfuhrunt.;inTheorieundAnwendungderLaplaceTransformation," Birkhauser,Stuttgart,1958. Gardner,M.F.,andJ.L.Barnes:"TransientsinLinearSystems,"vol.I, John Wiley&Sons,Inc.,NewYork,1942. Goldman,S.:"TransformationCalculus,"Prentice-Hall,Inc.,EnglewoodCliffs, N.J.,1949. Halperin,I.:"Introduction totheTheoryofDistributions,"UniversityofToronto Press,Toronto,1952. Holl,D.L.,C.G.Maple,andB.Vinograde:"IntroductiontotheLaplaceTrans- form,"Appleton-Century-Crofts,Inc.,NewYork,1959. Jury,E.I.: "SampledData Control Systems," John Wiley&Sons,Inc.,NewYork, 1958. Lago,G.V.,andD.L.Waidelich:"Transients inElectricalCircuits,"TheRonald PressCompany,New York,1958. Lighthill,M.J.:"An Introduction toFourierAnalysisandGeneralizedFunctions," CambridgeUniversity Press,NewYork,1958. McLachlan,N.W.:"ModernOperationalCalculus,withApplicationsinTechnical. Mathematics," St.Martin's Press,Inc.,NewYork,1948. Paley,R.,and N.Wiener:"Fourier Transforms in the ComplexDomain,"American Mathematical Society,Providence,R.I.,1934. Ragazzini,J.R.,andG.F.Franklin:"Sampled-dataControlSystems,"McGraw- HillBookCompany,Inc.,NewYork,1958. Schwartz, L.:"Theorie des distributions," vol.I,Herman &Cie,Paris,1950. Sneddon,I.N.:"FourierTransforms,"McGraw-HillBookCompany,Inc.,New York,1951. Titchmarsh,E.C.:"TheoryofFourierIntegrals,"OxfordUniversityPress,New York,1937. vanderPol,B.,andH.Bremmer:"OperationalCalculusBasedontheTwo-sided LaplaceIntegral,"CambridgeUniversityPress,NewYork,1950. Weber,E.:"LinearTransientAnalysis,"vols.I,II,JohnWiley&;Sons,Inc.,New York,1954-1956. Widder,D.V.:"TheLaplaceTransform,"PrincetonUniversityPress,Princeton, N.J.,1941. Wiener,N.:"TheFourierIntegral,"CambridgeUniversityPress,NewYork, 1933(DoverPublications,Inc.,NewYork,1958). INDEX Abscissa of convergence,one-sided Laplace integral,292,293 two-sidedLaplace integral,295 Algebraic singularity,185 Almost piecewise continuity,240 Amplitude modulation,applicationof convolution integral to,347 Analytic continuation,147 significance of,in definition of Laplace transform,298 Analytic function,32,152 Analytic geometry plane,24 Angle,of complex number, 21 initial, for sinusoidal wave,2 preservation of,by conformal mapping,75 Angle function,229 Are,differentiable,85 simple,85 Axis of convergence of Laplace integral, 293 Bessel function,46i Laplace transform of,466 Bilinear transformation,properties of, 70 Branch cut,171,185 Branch point,177 integration around,180 methods of locating,186 order of,173 Bromwich contour,forone-sided trans- form,320 fortwo-sided transform, 319 Cauchy integral formulae,106 Cauchy integral theorem,94-98 Cauchy principle of COl!vergence,for improper integrals,240 Cauchyprincipleofconvergence,for infinite series,117 Cauchy-Riemann differential equations, 34 Characteristic equation,389 Characteristic values,326,389 Complex number,angle of,21 exponential form,37 imaginary component of,20 imaginary part of,20 magnitude of,21 as an orderedpair,19 polar form,21 real component of,20 real part of,20 rectangular form,21 Complexplane,24 Conformal mapping,by analytic func- tion,73 preservation,of angles by,75 of shapes by,76 Conformal maps,bilinear function,70 exponential function,61 hyperboliccosine,62 reciprocal function,56,66 trigonometric sine and cosine,63 Conjugate,complex,4,23 Connected set,86 Connectivity, order of,87 of aregion,87 signific:1nceof,forintegrals,99 Continuity, definition of,28 Contour integral,89,92 Convergence(SIleImproper real inte- grals; Infinite series) Convolution in •plane,forone-sided transform,343 471 for two-sided transform, 349 Convolution integral,forFourier trans- form,338 472COMPLEXVARIABLESANDTHELAPLACETRANSFORM Convolutionintegral,forimpulse func- tion,428 forone-sidedLaplace transform,343 fortwo-sidedLaplace transform,340 Convolution theorem forZtransform, 458 Cosine integral function,468 Laplacetransform of,467 Current source,equivalent,405 Curve,simple closed,85 Deleted neighborhood,28 Derivative,definition forfunctionof a complex variable,29 fordiscontinuous functionof areal variable,367 of impulse function,417 of multivalued function,177 Distribution,433 Element of an analytic function,148 Entire function,153 Error function,468 Laplace transform of,467 Essential singularity,firstkind,142 second kind,143 Exponential form,37 Exponential function,36 Exponential integral function,468 Laplace transform of,467 Exponential order,functionof,287 Exponential type, function of,357 Extended plane,88 Field problems,solution in twodimen- sions,77-80 Final-value theorem,forLaplace transform,315 forZtransform,45J Finite plane,88 Forced response,390 Fourier integral,6,268 Fourier integral theorem,268-272 Fourierseries,5 Fourier transform,274 derivative of,275 symmetry of,276 Frequency,10 Frequency variable,10 generalized,12 Function, of acomplexvariable,24 analytic,32,152 entire,152 meromorphic,152 rational,152 transcendental,35 global definition of,151 of areal variable,almost piecewise continuity of,240 piecewise continuity of,235 Gain function,229 Gaussmean-valuetheorem,205 Global definition of afunction,151 Half plane of convergence forLaplace integral,293 Harmonic function,41 Heaviside expansion theorem,325 Helmholtztheorem,403 Norton's theorem as special case,405 Thevenin's theorem as spccial case, 404 Hilbert transform,225 Hyperbolicfunctions,38 Imaginary component ofcomplex number,20 Imaginary number,20 Imaginary part of complexnumber,20 Immittance function,398 combination of,400 self-immittance,400 transfer immittance,400 Improper real integrals,237,238 convergence of,237,239,240 absolute,239 uniform,248 Impulse function,414 higher-order,416,418 symbolic transform of,417,419 Impulse response,414 Impulsive response,415 Indexprinciple,211 Infinite series,116--122 convergence of,116,117 INDEX473 Infinite series, convergence, absolute,118 uniform,120 power(se6Power series) ratio test,119 root test,119 Infinity,point at,64 Initial-value theorem,forLaplace transform,315 forZtransform,451 Integral, contour, 89,92 upper bound,94 improper(S66Improper real integrals) line,90 real,theorems for,236 Integration,around branch points,180 over large circular arcs,254 by parts,236 by primitive functions,109 by residues,145 Inversionformula,forFourier trans- form,268,273 nonuniqueness of,for two-sided Laplace transform,315,321 forone-sidedLaplace transform,320 forone-sidedZtransform,452 for two-sidedLaplace transform,319 fortwo-sidedZtransform,459 uniquenessof,forFouriertransform and one-sidedLaplace transform, 276,320 Iterated integrals,finitelimits,242 infinite limits,247 Jordan curve,closed,86 Jordan curve theorem,88 Jordan's lemma,259 Laguerre polynomial,113,468 Laplace transform of,466 Laplace integral,convergence of,288, 289 one-sided,8,287 related toFourier integral,286 two-sided,286 Laplace transform,as an analytic func- tion,298 behavior at infinity,306 conditions to make anentire function, 309 Laplacetransform,derivative of,308 of derivative,311 of integral,312 inverse of(seeInversion formula) linear combination of,299 one-sided,9,287 sufficient conditions for,374 uniqueness of inverse,320 Laplace's equation,solution of,by func- tions of acomplex variable,77-80 in twodimension,41,77 Laurent series,134 expanded about asingularpoint,139 properties of coefficients forreal function,203 uniqueness of,136 Legendre polynomial,113,468 Laplacetransform of,466 Limit,definitionof,27 Line integral,90 Linear function,mapping properties of, 46 Lipshitz condition,271 Logarithm,definition,102 mapping properties of,176 Riemann surface,102 Logarithmic singularity,185 Mtest(seeWeierstrassMtest) Magnitude of complex number,21 Maximum,principle of,207 Meromorphic function,152 Minimum,principle of,207 Mittag-LetHer theorem,157 ModifiedZtransform,461 Morera's theorem,109 Multivalued function,expansion in series,188 Natural boundary,150 Natural mode,326,389 Natural response,389 Neighborhood,deleted,28 Nyquist criterion,213 Open region,86 Order,of branch point,142 of connectivity,87 474COMPLEXVARIABLESANDTHELAPLACETRANSFORM Order,of azero,212 Ordered pair,19 P&l'8Ilval'stheorem forFourier trans- form,279 Partial fractionexpansion,153 of meromorphic function,157 Partial sum of series,116 Periodic function,Laplace transform of, 436-438 response of system to, 438-442 Piecewise continuity, 235 Point set,86 Poisson's integrals, 215 transformed to imaginary axis,220 Pole,definition of,142 order of,142 of system functionrelated to natural response,389 Positive real function,208 Power series,125-129 circle of convergence,125 term-by-term differentiation,129 term-by-term integration,128 Primitive function,100 used in eyaluating integrals,109 Principal part of Laurent expansion, 142 Radim of convergenceforpower series, 126 Real component of complex number,20 Real function,201 Real number,20 Real part of complex number,20 Region,86 bounded,88 closed,87 connectivity of,87 open, 86 Regular point, 31 Residue,definition,144 formulas for,145 at infinity,146 Residue theorem,145 Resonance,391 Riemann sphere,64 Riemann surface,103,171 Riemann's theorem for trigonometric integrals,252-254 generalization of,354-355 Root locus,190-197 Roots,of complex numbers,22,37 of equations,212 Saddle point,186 Sampled transfer function,457 Sampling function,446 Series(8eeInfinite series) Set,connected,86 of measure zero,integration over,237 point,86 Shapes,preservation of,76 Shifting theorems,forLaplace trans- form,309,310 forZtransform,450 Sine integral function,468 Laplace transform of,467 Singular point,31 classification(seeAlgebraic singu- larity; Branch point; Essential singularity;Logarithmic singu- larity;Pole) at infinity,143 Singularity functions,424 Sinusoidal pulse,Laplace transform of, 304 Sinusoidal steady state,1-5,397 Stability,12,390 Strip of convergence for two-sided Laplace integral,296 Superposition,430 Symmetry ofFourier transforms,276 System function,2,406 angle function,229 gain function,229 relationships between real and imagi- nary parts,223-228 Taylor series,129-133 uniqueness of,132 Tchebysheff polynomial,199,468 theorem as special case of Helmholtz theorem,404 Transcendental functions,35 Transformation point due to afunction, 48 INDEX475 Transient response,389 Triangular pulse,Laplace transform of, 304 Trigonometrie funetions,38 Two-sided Laplace transform(Bee Laplace transform) Two-sided Ztransform,458 Uniform convergence,improper inte- grals,248 infinite series,121 Laplace integrals,289,291,294 Mtest, forintegrals,251 forseries,127 power series,127 Unit doublet,417 Unit impulse,414 Unit pulse,410 Unit ramp,424 Unit step,310 Voltage souree,equivalent,405 WeierstrassMtest,forintegrals,251 forseries,127 Ztransform,447 convolution theorem for,457 of functionmultiplied bye-a', 449 inversion formula for,451,459 modified,461 periodicproperties of,453 ofpowers of t,448 two-sided,458 Zerosof afunction,212 Wilbur R.LePage Complex Variables and the Laplace Transform for Engineers " ... anexcellenttext;thebestIhavefoundonthesubject. " - J.B.Sevart ,Dept.of MechanicalEngineering,U. of Wichita "An extremely useful textbook for both formal classes and for self-study . .... - SOCIETY FORINDUSTRIAL AND APPLIEDMATHEMATICS Engineers oftendonothavetimetotakeacoursein complexvariabletheoryasunder- graduates, yetitisone of the mostimportant and usefulbranches of mathematics, with many applications in engineering. This text is designed to remedy that need by supplying graduate engineering students (especially electrical engineering) with a course in the basic theory of complex variables, whichintum isessentialtothe understanding of transform theory.Presupposing agood knowledge of calculus, the book deals lucidly and rigorously with important mathematical concepts, striking an idealbalance between purely mathe- matical treatments that are too general for the engineer , and books of applied e n g i n ~ r i n g whichmay failtostress significantmathematical ideas. The text is divided into two basic parts: The first part (Chaps.1- 7) is devoted to the theory of complex variables and begins with an outline of the structure of system analysis and an explanation of basicmathematical and engineering terms.Chapter 2treats the founda - tion of the theory of a complex variable, centered around the Cauchy-Riemann equations. The nextthree chapters - conformal mapping, complex integration and infinite series - lead up to a particularly important chapter on multivalued functions, explaining the con- cepts of stability, branch points and Riemann surfaces.Numerous diagrams illustrate the physicalapplications of the mathematical conceptsinvolved. The second part (Chaps. 8- 16) covers Fourier and Laplace transform theory and some of its applications in engineering,beginning withachapter on realintegrals. Three impor- tantchaptersfollowontheFourierintegral ,theLaplaceintegral(one-sidedandtwo- sided) and convolutionintegrals. After achapter on additional properties of theLaplace integral , the book ends with four chapters (13- 16) on the application of transform theory to the solution of ordinary linear integrodifferential equations with constant coefficients, impulsefunctions ,periodic functionsand theincreasinglyimportantZ transform. Dr.Le Page's book isunique in its coverage of an unusually broad range of topics difficult to findin a single volume, while at the same time stressing fundamental concepts, careful attentiontodetailsandcorrectuseof terminology.Anextensiveselectionof interesting and valuable problems follows each chapter, and an excellent bibliography recommends furtherreading.Idealforhome study or asthe nucleus of a graduate course, this useful , practical and popular (8 printings in its hardcover edition) text offers students, engineers andresearchersacareful ,thoroughgroundinginthemathessentialtomanyareasof engineering. " ... an outstanding job ... "-AMERICANMATHEMATICALMONTHL¥ Unabridged, corrected republication of the McGraw-Hilledition,1961 .Numerousillus- trations.Bibliography_ Exercises. 483pp.5% x8y' .Paperbound.