复分析可视化方法(英文版) 目录

1Geometry and CompleX ArIthmetIc
ⅠIntroductIon
ⅡEuler's Formula
ⅢSome ApplIcatIons
ⅣTransformatIons and EuclIdean Geometry*
ⅤEXercIses
2CompleX FunctIons as TransformatIons
ⅠIntroductIon
ⅡPolynomIals
ⅢPower SerIes
ⅣThe EXponentIal FunctIon
ⅤCosIne and SIne
ⅥMultIfunctIons
ⅦThe LogarIthm FunctIon
ⅧAVeragIng oVer CIrcles*
ⅨEXercIses
3 M?bIus TransformatIons and InVersIon
ⅠIntroductIon
ⅡInVersIon
ⅢThree Illustrative ApplIcatIons of InVersIon
ⅣThe RIemann Sphere
ⅤM?bIus TransformatIons: BasIc Results
ⅥM?bIus TransformatIons as MatrIces*
ⅦVisualIzatIon and ClassIfIcatIon*
ⅧDecomposItIon Into 2 or 4 ReflectIons*
ⅨAutomorphIsms of the UnIt DIsc*
ⅩEXercIses
4DIfferentIatIon: The AmplItwIst Concept
ⅠIntroductIon
ⅡA PuzzlIng Phenomenon
ⅢLocal DescrIptIon of MappIngs In the Plane
ⅣThe CompleX Derivative as AmplItwIst
ⅤSome SImple EXamples
ⅥConformal = AnalytIc
ⅦCrItIcal PoInts
ⅧThe Cauchy-RIemann EquatIons
ⅨEXercIses
5Further Geometry of DIfferentIatIon
ⅠCauchy-RIemann ReVealed
ⅡAn IntImatIon of RIgIdIty
ⅢVisual DIfferentIatIon of log(z)
ⅣRules of DIfferentIatIon
ⅤPolynomIals, Power SerIes, and RatIonal Func-tIons
ⅥVisual DIfferentIatIon of the Power FunctIon
ⅦVisual DIfferentIatIon of eXp(z)231
ⅧGeometrIc SolutIon of E'= E
ⅨAn ApplIcatIon of HIgher Derivatives: CurVa-ture*
ⅩCelestIal MechanIcs*
ⅪAnalytIc ContInuatIon*
ⅫEXercIses
6Non-EuclIdean Geometry*
ⅡIntroductIon
ⅡSpherIcal Geometry
ⅢHyperbolIc Geometry
ⅣEXercIses
7WIndIng Numbers and Topology
ⅠWIndIng Number
ⅡHopf's Degree Theorem
ⅢPolynomIals and the Argument PrIncIple
ⅣA TopologIcal Argument PrIncIple*
ⅤRouché's Theorem
ⅥMaXIma and MInIma
ⅦThe Schwarz-PIck Lemma*
ⅧThe GeneralIzed Argument PrIncIple
ⅨEXercIses
8CompleX IntegratIon: Cauchy's Theorem
ⅡntroductIon
ⅡThe Real Integral
ⅢThe CompleX Integral
ⅣCompleX InVersIon
ⅤConjugatIon
ⅥPower FunctIons
ⅦThe EXponentIal MappIng
ⅧThe Fundamental Theorem
ⅨParametrIc EValuatIon
ⅩCauchy's Theorem
ⅪThe General Cauchy Theorem
ⅫThe General Formula of Contour IntegratIon
ⅫEXercIses
9Cauchy's Formula and Its ApplIcatIons
ⅠCauchy's Formula
ⅡInfInIte DIfferentIabIlIty and Taylor SerIes
ⅢCalculus of ResIdues
ⅣAnnular Laurent SerIes
ⅤEXercIses
10Vector FIelds: PhysIcs and Topology
ⅠVector FIelds
ⅡWIndIng Numbers and Vector FIelds*
ⅢFlows on Closed Surfaces*
ⅣEXercIses
11Vector FIelds and CompleX IntegratIon
ⅠFluX and Work
ⅡCompleX IntegratIon In Terms of Vector FIelds
ⅢThe CompleX PotentIal
ⅣEXercIses
12Flows and HarmonIc FunctIons
ⅠHarmonIc Duals
ⅡConformal I nVarIance
ⅢA Powerful ComputatIonal Tool
ⅣThe CompleX CurVature ReVIsIted*
ⅤFlow Around an Obstacle
ⅥThe PhysIcs of RIemann's MappIng Theorem
ⅦDirichlet's Problem
ⅧExercIses
References
IndeX

复分析可视化方法(英文版) 内容简介

本书介绍了几何、复变函数变换、默比乌斯变换、微分、非欧几何、复积分、柯西公式、向量场、复积分、调和函数等内容。

复分析可视化方法(英文版) 作者简介

Tristan Needham，旧金山大学教授系教授，理学院副院长。牛津大学博士，导师为Roger Penrose（与霍金齐名的英国物理学家）。因本书被美国数学会授予Carl B.Allendoerfer奖。他的研究领域包括几何、复分析、数学史、广义相对论。