• 复分析
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复分析

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作者Elias、Rami Shakarchi 著

出版社世界图书出版公司

出版时间2012-12

版次1

装帧平装

货号A1

上书时间2024-12-23

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图书标准信息
  • 作者 Elias、Rami Shakarchi 著
  • 出版社 世界图书出版公司
  • 出版时间 2012-12
  • 版次 1
  • ISBN 9787510040542
  • 定价 59.00元
  • 装帧 平装
  • 开本 16开
  • 纸张 胶版纸
  • 页数 379页
  • 正文语种 英语
【内容简介】
  《复分析》是一部为数学及相关专业大学二年级和三年级学生编写的教材,理论与实践并重。为了便于非数学专业的学生学习,全书内容简明、易懂,读者只需掌握微积分和线性代数知识。与《复分析》相配套的教材《傅立叶分析导论》和《实分析》也已影印出版。《复分析》已被哈佛大学和加利福尼亚理工学院选为教材。目次:复分析预备知识;柯西定理及其应用;亚纯函数和对数;傅立叶转换;Gamma和Zeta函数;Zeta函数和素数定理;共形映射;椭圆函数;Theta函数。
【目录】
foreword
introduction
chapter1.preliminariestocomplexanalysis
1complexnumbersandthecomplexplane
1.1basicproperties
1.2convergence
1.3setsinthecomplexplane
2functionsonthecomplexplane
2.1continuousfunctions
2.2holomorphicfunctions
2.3powerseries
3integrationalongcurves
4exercises

chapter2.cauchy'stheoremanditsapplications
1goursat'stheorem
2localexistenceofprimitivesandcauchy'stheoreminadisc
3evaluationofsomeintegrals
4cauchy'sintegralformulas
5furtherapplications
.5.1morera'stheorem
5.2sequencesofholomorphicfunctions
5.3holomorphicfunctionsdefinedintermsofintegrals
5.4schwarzreflectionprinciple
5.5runge'sapproximationtheorem
6exercises
7problems

chapter3.meromorphicfunctionsandthelogarithm
1zerosandpoles
2theresidueformula
2.1examples
3singularitiesandmeromorphicfunctions
4theargumentprincipleandapplications
5homotopiesandsimplyconnecteddomains
6thecomplexlogarithm
7fourierseriesandharmonicfunctions
8exercises
9problems

chapter4.thefouriertransform
1theclass
2actionofthefouriertransformon
3paley-wienertheorem
4exercises
5problems

chapter5.entirefunctions
1jensen'sformula
2functionsoffiniteorder
3infiniteproducts
3.1generalities
3.2example:theproductformulaforthesinefunction
4weierstrassinfiniteproducts
5hadamard'sfactorizationtheorem
6exercises
7problems

chapter6.thegammaandzetafunctions
1thegammafunction
1.1analyticcontinuation
1.2furtherpropertiesofγ
2thezetafunction
2.1functionalequationandanalyticcontinuation
3exercises
4problems

chapter7.thezetafunctionandprimenumberthe-orem
1zerosofthezetafunction
1.1estimatesfor1/ζ(s)
2reductiontothefunctionsψandψ1
2.1proofoftheasymptoticsforψ1
noteoninterchangingdoublesums
3exercises
4problems

chapter8.conformalmappings
iconformalequivalenceandexamples
1.1thediscandupperhalf-plane
1.2furtherexamples
1.3thedirichletprobleminastrip
2theschwarzlemma;automorphismsofthediscandupperhalf-plane
2.1automorphismsofthedisc
2.2automorphismsoftheupperhalf-plane
3theriemannmappingtheorem
3.1necessaryconditionsandstatementofthetheorem
3.2montel'stheorem
3.3proofoftheriemannmappingtheorem
4conformalmappingsontopolygons
4.1someexamples
4.2theschwarz-christoffelintegral
4.3boundarybehavior
4.4themappingformula
4.5returntoellipticintegrals
5exercises
6problems

chapter9.anintroductiontoellipticfunctions
1ellipticfunctions
1.1liouville'stheorems
1.2theweierstrasspfunction
2themodularcharacterofellipticfunctionsandeisensteinseries
2.1eisensteinseries
2.2eisensteinseriesanddivisorfunctions
3exercises
4problems

chapter10.applicationsofthetafunctions
1productformulaforthejacobithetafunction
1.1furthertransformationlaws
2generatingfunctions
3thetheoremsaboutsumsofsquares
3.1thetwo-squarestheorem
3.2thefour-squarestheorem
4exercises
5problems
appendixa:asymptotics
ibesselfunctions
2laplace'smethod;stirling'sformula
3theairyfunction
4thepartitionfunction
5problems
appendixb:simpleconnectivityandjordancurvetheorem
1equivalentformulationsofsimpleconnectivity
2thejordancurvetheorem
2.1proofofageneralformofcauchy'stheorem
notesandreferences
bibliography
symbolglossary
index
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