Complex Analysis: Theory and Applications

423 Pages·2019·5.239 MB·English

by  Teodor Bulboaca, Santosh B. Joshi, Pranay Goswami

TeodorBulboacǎ,SantoshB.Joshi,andPranayGoswami
ComplexAnalysis
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Teodor Bulboacǎ, Santosh B. Joshi, and
Pranay Goswami
Complex Analysis
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Theory and Applications
MathematicsSubjectClassification2010
30-XX
Authors
Prof.Dr.Habil.TeodorBulboacǎ
Babeş-BolyaiUniversity
FacultyofMathematicsandComputerScience
Str.MihailKogălniceanuNr.1
400084Cluj-Napoca
Romania
[email protected]
Prof.Dr.SantoshB.Joshi
WalchandCollegeofEngineering
DepartmentofMathematics
416415Maharastra
India
[email protected]
Prof.Dr.PranayGoswami
AmbedkarUniversityDelhi
KashmereGateCampus
LothianRoad,KashmereGate
110006Delhi
India
[email protected]
ISBN978-3-11-065782-1
e-ISBN(PDF)978-3-11-065786-9
e-ISBN(EPUB)978-3-11-065803-3
LibraryofCongressControlNumber:2019938423
BibliographicinformationpublishedbytheDeutscheNationalbibliothek
TheDeutscheNationalbibliothekliststhispublicationintheDeutscheNationalbibliografie;
detailedbibliographicdataareavailableontheInternetathttp://dnb.dnb.de.
©2019WalterdeGruyterGmbH,Berlin/Boston
Coverimage:Lokal_Profil(Grafic»conformal_grid«)–MichaelBernhart(surface)
Typesetting:VTeXUAB,Lithuania
Printingandbinding:CPIbooksGmbH,Leck
www.degruyter.com
Contents
Preface|XI
1 Complexnumbers|1
1.1 Thefieldofthecomplexnumbers|1
1.2 Thecomplexplane|3
1.3 Thetopologicalandmetricstructureofthecomplexplane|6
1.3.1 Basicdefinitionsandnotation|6
1.4 Complexfunction,limits,continuity|9
1.5 Thecompactifiedcomplexplane|10
1.5.1 Thegeometricinterpretationoftheφfunction|12
ℂ̂
1.5.2 Thetopologicalstructureof |12
1.6 Exercises|14
2 Holomorphicfunctions|17
2.1 Thederivativeoftherealvaluedcomplexfunctions|17
2.2 Thedifferentiabilityofacomplexfunction|19
2.3 Thederivativeofacomplexfunction|23
2.3.1 Thepropertiesofthederivative|25
2.4 Thegeometricinterpretationofthederivative|29
2.5 Entirefunctions|33
2.5.1 Thepolynomialfunction|33
2.5.2 Theexponentialfunction|33
2.5.3 Complextrigonometricfunctions|34
2.5.4 Complexhyperbolicfunctions|35
2.6 Bilineartransforms|35
2.6.1 Decompositionsinelementaryfunctions|36
2.7 TheMöbius-typegroups|39
2.8 Multivaluedfunctions|43
2.8.1 Thelogarithmicfunction|43
2.8.2 Inversetrigonometricfunctions|45
2.8.3 Thepowerfunction|46
2.9 Exercises|47
2.9.1 Realvariablecomplexfunctions|47
2.9.2 Thederivativeofacomplexfunction|47
2.9.3 Entirefunctions|49
2.9.4 Bilineartransforms|49
3 Thecomplexintegration|51
VI | Contents
3.1 Thehomotopictheoryofthepaths|51
3.1.1 Simplyconnecteddomains|55
3.1.2 Functionsofboundedvariationandpaths|58
3.2 Thecomplexintegral|60
3.2.1 TheRiemann–Stieltjesintegralforcomplexvaluedfunctions|60
3.3 TheCauchytheorem|66
3.3.1 Theconnectionbetweentheintegralandtheprimitivefunction|66
3.3.2 TheCauchytheorem|74
3.4 TheCauchyformulaforthedisc|79
3.5 Theanalyticalbranchesofmultivaluedfunctions|82
3.6 Theindexofapath(curve)withrespecttoapoint|84
3.7 Cauchyformulaforclosedcurves|87
3.8 SomeconsequencesofCauchyformula|88
3.9 SchwarzandPoissonformulas|91
3.10 Exercises|94
3.10.1 Thecomplexintegral|94
3.10.2 TheCauchytheorem|96
3.10.3 TheCauchyformulaforthedisc|97
3.10.4 SomeconsequencesofCauchyformula|101
3.10.5 Multivaluedfunctionsanalyticalbranches|102
4 Sequencesandseriesofholomorphicfunctions|105
4.1 Sequencesofholomorphicfunctions|105
4.2 Seriesoffunctions|107
4.3 Powerseries|108
4.4 Theanalyticityofholomorphicfunctions|111
4.5 Thezerosofholomorphicfunctions|114
4.6 Themaximumprincipleoftheholomorphicfunctions|117
4.7 Laurentseries|122
4.8 Isolatedsingularpoints|127
4.9 Meromorphicfunctions|131
4.10 Exercises|134
4.10.1 Powerseries|134
4.10.2 TaylorandLaurentseries|135
4.10.3 Isolatedsingularpoints|138
4.10.4 Themodulemaximumoftheholomorphicfunctions|139
5 Residuetheory|141
5.1 Residuetheorem|141
5.2 Applicationsoftheresiduetheoremtothecalculationofthe
integrals|144
5.3 Thestudyofmeromorphicfunctionswiththeresiduetheorem|158
Contents | VII
5.4 Exercises|165
5.4.1 Residuetheorem|165
5.4.2 Applicationsoftheresiduetheoremtothecalculationofthe
trigonometricintegrals|168
5.4.3 Applicationsoftheresiduetheoremtothecalculationoftheimproper
integrals|169
5.4.4 Thestudyofmeromorphicfunctionsusingtheresiduetheorem|169
6 Conformalrepresentations|171
6.1 Specialclassesofholomorphicfunctions|171
6.2 Univalentfunctions|175
6.3 Theproblemofconformalrepresentation|179
6.4 TheRiemannmappingtheorem|181
6.5 Exercises|187
7 Solutionstothechapterwiseexercises|191
7.1 SolutionstotheexercisesofChapter1|191
7.2 SolutionstotheexercisesofChapter2|199
7.3 SolutionstotheexercisesofChapter3|218
7.4 SolutionstotheexercisesofChapter4|277
7.5 SolutionstotheexercisesofChapter5|322
7.6 SolutionstotheexercisesofChapter6|380
Bibliography|405
Index|407

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DedicatedtothememoryofProfessorPetruT.Mocanu(1931–2016)