Mathematical Methods for Engineers and Scientists 1: Complex Analysis and Linear Algebra

498 Pages·2022·6.369 MB·English

by  Kwong-Tin Tang

Kwong-Tin Tang
Mathematical
Methods
for Engineers
and Scientists 1
Complex Analysis and Linear Algebra
Second Edition
Mathematical Methods for Engineers
and Scientists 1
Kwong-Tin Tang
Mathematical Methods
for Engineers and Scientists 1
Complex Analysis and Linear Algebra
Second Edition
Kwong-TinTang
DepartmentofPhysics
PacificLutheranUniversity
Tacoma,WA,USA
ISBN 978-3-031-05677-2 ISBN 978-3-031-05678-9 (eBook)
https://doi.org/10.1007/978-3-031-05678-9
1stedition:©Springer-VerlagBerlinHeidelberg2007
2ndedition:©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringer
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Inlovingmemoryofmyparents
Preface to Second Edition
Therearetwomorechaptersinthisedition.
Theoneonconformalmappingisbyreaders’request.Hopefullythismeansthat
thesubjectisimportantandmyleisurelyapproachisappealing.Imustconfessthat
Itaughtthesematerialsonlyoncebefore.Theyaremostlymypersonalnotes.
Therewerethreechaptersondeterminantsandmatrices.Togethertheyformthe
bodyoflinearalgebra.However,Ifounditeasytogetlostinthedetailsofcalculations.
So I wrote a summary of the computation aspects and gave a simple geometrical
description of linear algebra in terms of vector space. These are the contents of
Chap. 5. Because of the time line they were written, there are some overlaps with
the following chapters. Since pedagogically a certain amount of repetition is not
necessarilybad,Ididnotremovethem.
Iwanttothankallthosewhopointedoutthemistakesofthefirstedition.Especially
IwanttothankmycolleaguesProfessorRichardLouiewhogavemealistoferrors,
and Prof. Chang-li Yiu who wrote part of the vector space. I must also thank my
wife,Pauline,whohastakenupallhousechores.OnceagainIwanttothankMattew
Hackerforhishelpwithcomputerproblems.
May this set of books be an acceptable offering in the eyes of my Lord Jesus
Christ.
Tacoma,Washington Kwong-TinTang
March2022
vii
Preface to the First Edition
For thirty-some years, I have taught two “Mathematical Physics” courses. One of
themwaspreviouslynamed“EngineeringAnalysis”.Thereareseveraltextbooksof
unquestionablemeritforsuchcourses,butIcouldnotfindonethatfittedourneeds.
Itseemedtomethatstudentsmighthaveaneasiertimeifsomechangesweremade
inthesebooks.Iendedupusingclassnotes.ActuallyIfeltthesameaboutmyown
notes,sotheygotchangedagainandagain.Throughouttheyears,manystudentsand
colleagues haveurgedmetopublishthem.Iresisteduntilnow,becausethetopics
were not new and I was not sure that my way of presenting them was really that
muchbetterthanothers.Inrecentyears,someformerstudentscamebacktotellme
thattheystillfoundmynotesusefulandlookedatthemfromtimetotime.Thefact
thattheyalwayssingledoutthesecourses,amongmanyothersIhavetaught,made
methinkthatbesidesbeingkind,theymightevenmeanit.Perhapsitisworthwhile
tosharethesenoteswithawideraudience.
It took far more work than expected to transcribe the lecture notes into printed
pages. The notes were written in an abbreviated way without much explanation
betweenanytwoequations,becauseIwassupposedtosupplythemissinglinksin
person.HowmuchdetailIwouldgointodependedonthereactionofthestudents.
Nowwithouttheminfrontofme,Ihadtodecidetheappropriateamountofderivation
tobeincluded.Ichosetoerronthesideoftoomuchdetailratherthantoolittle.As
aresult,thederivationdoesnotlookveryelegant,butIalsohopeitdoesnotleave
anygapinstudents’comprehension.
PreciselystatedandelegantlyprovedtheoremslookedgreattomewhenIwasa
young faculty member. But in later years, I found that elegance in the eyes of the
teachermightbestumblingblocksforstudents.NowIamconvincedthatbeforethe
student can use a mathematical theorem with confidence, he must first develop an
intuitivefeeling.Themosteffectivewaytodothatistofollowasufficientnumber
ofexamples.
Thisbookiswrittenforstudentswhowanttolearnbutneedafirmhand-holding.I
hopetheywillfindthebookreadableandeasytolearnfrom.Learning,asalways,has
tobedonebythestudentherselforhimself.Noonecanacquiremathematicalskill
withoutdoingproblems,themorethebetter.However,realisticallystudentshavea
ix
x PrefacetotheFirstEdition
finiteamountoftime.Theywillbeoverwhelmedifproblemsaretoonumerous,and
frustratedifproblemsaretoodifficult.Acommonpracticeintextbooksistolistalarge
numberofproblemsandlettheinstructortochooseafewforassignments.Itseems
to me that is not a confidence building strategy. A self-learning person would not
knowwhattochoose.Thereforeamoderatenumberofnotoverlydifficultproblems,
with answers, are selected at the end of each chapter. Hopefully after the student
hassuccessfullysolvedallofthem,hewillbeencouragedtoseekmorechallenging
ones.Thereareplentyofproblemsinotherbooks.Ofcourse,aninstructorcanalways
assignmoreproblemsatlevelssuitabletotheclass.
Oncertaintopics,Iwentfartherthanmostothersimilarbooks,notinthesenseof
esotericsophistication,butinmakingsurethatthestudentcancarryouttheactual
calculation.Forexample,thediagonalizationofadegenerateHermitianmatrixisof
considerableimportanceinmanyfields.Yettomakeitclearinasuccinctwayisnot
easy.Iusedseveralpagestogiveadetailedexplanationofaspecificexample.
Professor I.I. Rabi used to say “All textbooks are written with the principle of
least astonishment”. Well, there is a good reason for that. After all, textbooks are
supposedtoexplainawaythemysteriesandmaketheprofoundobvious.Thisbook
isnoexception.Nevertheless,Istillhopethereaderwillfindsomethinginthisbook
exciting.
This volume consists of three chapters on complex analysis and three chapters
on theory of matrices. In subsequent volumes, we will discuss vector and tensor
analysis, ordinary differential equations and Laplace transforms, Fourier analysis
andpartialdifferentialequations.Studentsaresupposedtohavealreadycompleted
twoorthreesemestersofcalculusandayearofcollegephysics.
This book is dedicated to my students. I want to thank my A students and B
students,theirdiligenceandenthusiasmhavemadeteachingenjoyableandworth-
while.IwanttothankmyCstudentsandDstudents,theirdifficultiesandmistakes
mademesearchforbetterexplanations.
IwanttothankBradOrawfordrawingmanyfiguresinthisbook,andMatthew
Hackerforhelpingmetotypesetthemanuscript.
IwanttoexpressmydeepestgratitudetoProfessorS.H.Patil,IndianInstituteof
Technology,Bombay.Hehasreadtheentiremanuscriptandprovidedmanyexcellent
suggestions. He has also checked the equations and the problems and corrected
numerouserrors.Withouthishelpandencouragement,Idoubtthereisthisbook.
Theresponsibilityforremainingerrorsis,ofcourse,entirelymine.Iwillgreatly
appreciateiftheyarebroughttomyattention.
Tacoma,Washington K.TTang
October2005
Contents
PartI ComplexAnalysis
1 ComplexNumbers ............................................. 3
1.1 OurNumberSystem ........................................ 3
1.1.1 AdditionandMultiplicationofIntegers ................. 4
1.1.2 InverseOperations ................................... 5
1.1.3 NegativeNumbers ................................... 6
1.1.4 FractionalNumbers .................................. 7
1.1.5 IrrationalNumbers ................................... 8
1.1.6 ImaginaryNumbers .................................. 9
1.2 Logarithm ................................................. 13
1.2.1 Napier’sIdeaofLogarithm ............................ 13
1.2.2 Briggs’CommonLogarithm ........................... 15
1.3 APeculiarNumberCallede ................................. 18
1.3.1 TheUniquePropertyofe ............................. 18
1.3.2 TheNaturalLogarithm ............................... 19
1.3.3 ApproximateValueofe ............................... 21
1.4 TheExponentialFunctionasanInfiniteSeries .................. 22
1.4.1 CompoundInterest ................................... 22
1.4.2 TheLimitingProcessRepresentinge ................... 24
1.4.3 TheExponentialFunctionex .......................... 24
1.5 UnificationofAlgebraandGeometry .......................... 25
1.5.1 TheRemarkableEulerFormula ........................ 25
1.5.2 TheComplexPlane .................................. 26
1.6 PolarFormofComplexNumbers ............................. 29
1.6.1 PowersandRootsofComplexNumbers ................. 31
1.6.2 TrigonometryandComplexNumbers ................... 34
1.6.3 GeometryandComplexNumbers ...................... 41
1.7 ElementaryFunctionsofComplexVariable .................... 48
1.7.1 ExponentialandTrigonometricFunctionsofz ........... 48
1.7.2 HyperbolicFunctionsofz ............................. 50
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xii Contents
1.7.3 LogarithmandGeneralPowerofz ..................... 52
1.7.4 InverseTrigonomericandHyperbolicFunctions .......... 58
Exercises ....................................................... 61
2 ComplexFunctions ............................................. 65
2.1 AnalyticFunctions ......................................... 65
2.1.1 ComplexFunctionasMappingOperation ............... 66
2.1.2 DifferentiationofaComplexFunction .................. 67
2.1.3 Cauchy-RiemannConditions .......................... 69
2.1.4 Cauchy-RiemannEquationsinPolarCoordinates ......... 72
2.1.5 AnalyticFunctionasaFunctionofzAlone .............. 74
2.1.6 AnalyticFunctionandLaplace’sEquation ............... 79
2.2 ComplexIntegration ........................................ 87
2.2.1 LineIntegralofaComplexFunction .................... 87
2.2.2 ParametricFormofComplexLineIntegral .............. 89
2.3 Cauchy’sIntegralTheorem .................................. 92
2.3.1 Green’sLemma ...................................... 92
2.3.2 Cauchy-GoursatTheorem ............................. 94
2.3.3 FundamentalTheoremofCalculus ..................... 96
2.4 ConsequencesofCauchy’sTheorem .......................... 99
2.4.1 PrincipleofDeformationofContours ................... 99
2.4.2 TheCauchyIntegralFormula .......................... 100
2.4.3 DerivativesofAnalyticFunction ....................... 102
Exercises ....................................................... 109
3 ComplexSeriesandTheoryofResidues .......................... 115
3.1 ABasicGeometricSeries .................................... 115
3.2 TaylorSeries ............................................... 116
3.2.1 TheComplexTaylorSeries ............................ 116
3.2.2 ConvergenceofTaylorSeries .......................... 117
3.2.3 AnalyticContinuation ................................ 119
3.2.4 UniquenessofTaylorSeries ........................... 121
3.3 LaurentSeries ............................................. 125
3.3.1 UniquenessofLaurentSeries .......................... 129
3.4 TheoryofResidues ......................................... 136
3.4.1 ZerosandPoles ...................................... 136
3.4.2 DefinitionoftheResidue .............................. 137
3.4.3 MethodsofFindingResidues .......................... 138
3.4.4 Cauchy’sResidueTheorem ............................ 142
3.4.5 SecondResidueTheorem ............................. 143
3.5 EvaluationofRealIntegralswithResidues ..................... 150
3.5.1 IntegralsofTrigonometricFunctions .................... 150
3.5.2 Improper Integrals I: Closing the Contour
withaSemicircleatInfinity ........................... 154
3.5.3 FourierIntegralandJordan’sLemma ................... 157