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 Copyright1961byWilburR.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 originallypublishedin1961byMcGrawHill,Inc. InternationalStandardBookNumber:0486-639266 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+ (32 - 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 B468 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=Aeia (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+ ClJ2R2C2 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 Vo, 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+ifO)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+ifO)'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 Vb(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)eia(.. ) 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' insteadofeiw1 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)eiwt + 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 ofeiw"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=(AI,A2)(2-1) A second complexnumber may be designated by B=(BI,B2) Using these as examples, the algebraic operations are defined as follows: 1.Identity A=B if and only if Al = BI and A, =B2 19 (2-2) 20COMPLEXVARIABLESANDTHELAPLACETRANSFORM 2.Addition (2-3) 3.Multiplication AB =(AlBl- A2B"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)=(-A2,A1) 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=(A1,A2) =(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 angB =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 Analyticgeometry 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,since82 has the meaning (8)(8),and multiplication has been defined.Thus, foreach point in the 8plane or w=(0- + jw)(o- + jw) =0-2 - w2 + j20-w U=0-2 - w2 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=81 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+a2B2 ++an8" w=-Q(-s)='bo'---+7----;b--=IS-+--i--cb,.: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 (u2 + 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 u2 - 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,andwithallanglelabelschangedbyr (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=82 beingoneexample.Here themutually perpen-dicular real and imaginary axes in the s plane transform into lines in the w plane intersecting at 1800,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 limAAw = 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)=Aeia,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 =jp1-" f ~ "[cos(1- n)4>+ jsin(1- n),p]d,p INTEGRATION105 whichiszerowhenn1andconfirms Eq.(4-23)when n=1.Thus, (d8=0n1(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,letC1 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 0c \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 If(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 istheradiusofC1Furthermore,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 < Ro Also,if and so,by Theorem 5-4, Is- sollalells- 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 .. andlaklTo 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 Ro,the radius ofconvergence,and sowecan state the followingtheorem: Theorem5-5.Afunctionrepresentedbyapowerseriesexpanded about a point So,with radius of convergence Ro, 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+82+ 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 Ro,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 Ro, 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)82 - 38+2 expanded about the point 80= O.Wecan perform adivision algorithm as follows: +%8+ %82 2- 38+8211 1 - %8+ %8-%8- %82 +%83 %82 - %83 %82 - 2H83 +%84 1%83 - %84 For the finitenumber ofsteps shown, f(8)=+%8+%82 + wherethequantityinparenthesesistheremainderterm.Without carryingoutthedetails,itisapparentthat forsmall181the remainder approaches zero as the algorithm iscontinued; the series +%8+%82 +lX683+... convergestof(8)for1812 882 83 84 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 onthesequenceofcurvesC1,C2, 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 2181 2'8' c082,-1-2f+4f+ obtain series expansions for (a)Sinl8(b)COSl8 (HINT:Consider term-by-term differentiation.) 1-14.Given the power series .8'8' SIn8 "*8-3i+5i 81 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"nla3 =~kl{n- k)! .. -I: 165 whichconvergesforR,< \8\< RtDetermineitsregionofuniform convergence. 6-20.For the function 82 + 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, 82 (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' 83 - 282 + 8 (e)88+ 38" 88 - 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-'-e21 (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 sin28 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 10, what is the above integral if the contour is changed(1)toC1 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)CA,aunit circle 'centered at 8=0 (b)Cb,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=In-! (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=x2 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 -=x2 -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=82 (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+ pei';w=1 + rei9 and1 + 2rei'+ rZei26 =1 + pei'; 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 w2 =(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 82 =(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