Applied Complex Variables for Scientists and Engineers

452 Pages·2010·3.856 MB·English

by  Yue Kuen Kwok

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AppliedComplexVariablesforScientistsandEngineers
SecondEdition
YueKuenKwok

Applied Complex Variables for
Scientists and Engineers
Second Edition
YueKuenKwok
HongKongUniversityofScienceandTechnology
CAMBRIDGE UNIVERSITY PRESS
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Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521701389
© Y. K. Kwok 2010
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2010
ISBN-13 978-0-511-77500-0 eBook (EBL)
ISBN-13 978-0-521-70138-9 Paperback
Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
Contents
Preface pageix
1 ComplexNumbers 1
1.1 Complexnumbersandtheirrepresentations 1
1.2 Algebraicpropertiesofcomplexnumbers 4
1.2.1 DeMoivre’stheorem 7
1.3 Geometricpropertiesofcomplexnumbers 13
1.3.1 nthrootsofunity 16
1.3.2 Symmetrywithrespecttoacircle 17
1.4 Sometopologicaldefinitions 23
1.5 ComplexinfinityandtheRiemannsphere 29
1.5.1 TheRiemannsphereandstereographicprojection 30
1.6 Applicationstoelectricalcircuits 33
1.7 Problems 36
2 AnalyticFunctions 46
2.1 Functionsofacomplexvariable 46
2.1.1 Velocityoffluidflowemanatingfromasource 48
2.1.2 Mappingpropertiesofcomplexfunctions 50
2.1.3 Definitionsoftheexponentialandtrigonometric
functions 53
2.2 Limitandcontinuityofcomplexfunctions 54
2.2.1 Limitofacomplexfunction 54
2.2.2 Continuityofacomplexfunction 58
2.3 Differentiationofcomplexfunctions 61
2.3.1 Complexvelocityandacceleration 63
2.4 Cauchy–Riemannrelations 64
2.4.1 Conjugatecomplexvariables 69
v
vi Contents
2.5 Analyticity 70
2.6 Harmonicfunctions 74
2.6.1 Harmonicconjugate 75
2.6.2 Steadystatetemperaturedistribution 80
2.6.3 Poisson’sequation 84
2.7 Problems 85
3 Exponential,LogarithmicandTrigonometricFunctions 93
3.1 Exponentialfunctions 93
3.1.1 Definitionfromthefirstprinciples 94
3.1.2 Mappingpropertiesofthecomplexexponential
function 97
3.2 Trigonometricandhyperbolicfunctions 97
3.2.1 Mappingpropertiesofthecomplexsinefunction 102
3.3 Logarithmicfunctions 104
3.3.1 Heatsource 106
3.3.2 Temperaturedistributionintheupperhalf-plane 108
3.4 Inversetrigonometricandhyperbolicfunctions 111
3.5 Generalizedexponential,logarithmic,andpower
functions 115
3.6 Branchpoints,branchcutsandRiemannsurfaces 118
3.6.1 Joukowskimapping 123
3.7 Problems 126
4 ComplexIntegration 133
4.1 Formulationsofcomplexintegration 133
4.1.1 Definiteintegralofacomplex-valuedfunctionofa
realvariable 134
4.1.2 Complexintegralsaslineintegrals 135
4.2 Cauchyintegraltheorem 142
4.3 Cauchyintegralformulaanditsconsequences 151
4.3.1 Derivativesofcontourintegrals 153
4.3.2 Morera’stheorem 157
4.3.3 ConsequencesoftheCauchyintegralformula 158
4.4 Potentialfunctionsofconservativefields 162
4.4.1 Velocitypotentialandstreamfunctionoffluid
flows 162
4.4.2 Electrostaticfields 175
4.4.3 Gravitationalfields 179
4.5 Problems 183
Contents vii
5 TaylorandLaurentSeries 194
5.1 Complexsequencesandseries 194
5.1.1 Convergenceofcomplexsequences 194
5.1.2 Infiniteseriesofcomplexnumbers 196
5.1.3 Convergencetestsofcomplexseries 197
5.2 Sequencesandseriesofcomplexfunctions 200
5.2.1 Convergenceofseriesofcomplexfunctions 201
5.2.2 Powerseries 206
5.3 Taylorseries 215
5.4 Laurentseries 221
5.4.1 Potentialflowpastanobstacle 230
5.5 Analyticcontinuation 233
5.5.1 Reflectionprinciple 236
5.6 Problems 238
6 SingularitiesandCalculusofResidues 248
6.1 Classificationofsingularpoints 248
6.2 ResiduesandtheResidueTheorem 255
6.2.1 Computationalformulasforevaluatingresidues 257
6.3 Evaluationofrealintegralsbyresiduecalculus 268
6.3.1 Integralsoftrigonometricfunctionsover[0,2π] 268
6.3.2 Integralsofrationalfunctions 269
6.3.3 Integralsinvolvingmulti-valuedfunctions 271
6.3.4 Miscellaneoustypesofintegral 275
6.4 Fouriertransforms 278
6.4.1 Fourierinversionformula 279
6.4.2 EvaluationofFourierintegrals 285
6.5 Cauchyprincipalvalueofanimproperintegral 288
6.6 Hydrodynamicsinpotentialfluidflows 295
6.6.1 Blasiuslawsofhydrodynamicforceandmoment 295
6.6.2 Kutta–Joukowski’sliftingforcetheorem 299
6.7 Problems 300
7 BoundaryValueProblemsandInitial-Boundary
ValueProblems 311
7.1 Integralformulasofharmonicfunctions 312
7.1.1 Poissonintegralformula 312
7.1.2 Schwarzintegralformula 319
7.1.3 Neumannproblems 324
7.2 TheLaplacetransformanditsinversion 326
7.2.1 Bromwichintegrals 330
viii Contents
7.3 Initial-boundaryvalueproblems 336
7.3.1 Heatconduction 337
7.3.2 Longitudinaloscillationsofanelasticthinrod 341
7.4 Problems 346
8 ConformalMappingsandApplications 358
8.1 Conformalmappings 358
8.1.1 InvarianceoftheLaplaceequation 364
8.1.2 Hodographtransformations 372
8.2 Bilineartransformations 375
8.2.1 Circle-preservingproperty 378
8.2.2 Symmetry-preservingproperty 381
8.2.3 Somespecialbilineartransformations 390
8.3 Schwarz–Christoffeltransformations 399
8.4 Problems 409
AnswerstoProblems 419
Index 434