| 本书是复分析领域近年来较有影响的一本著作。作者用丰富的图例展示各种概念、定理和证明思路，十分便于读者理解，充分揭示了复分析的数学之美。书中讲述的内容有几何、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。 |
| 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。 |
| 1 Geometry and CompleX ArIthmetIc Ⅰ IntroductIon Ⅱ Euler's Formula Ⅲ Some ApplIcatIons Ⅳ TransformatIons and EuclIdean Geometry* Ⅴ EXercIses 2 CompleX FunctIons as TransformatIons Ⅰ IntroductIon Ⅱ PolynomIals Ⅲ Power SerIes Ⅳ The EXponentIal FunctIon Ⅴ CosIne and SIne Ⅵ MultIfunctIons Ⅶ The LogarIthm FunctIon Ⅷ AVeragIng oVer CIrcles* Ⅸ EXercIses 3 M?bIus TransformatIons and InVersIon Ⅰ IntroductIon Ⅱ InVersIon Ⅲ Three Illustrative ApplIcatIons of InVersIon Ⅳ The RIemann Sphere Ⅴ M?bIus TransformatIons: BasIc Results Ⅵ M?bIus TransformatIons as MatrIces* Ⅶ VisualIzatIon and ClassIfIcatIon* Ⅷ DecomposItIon Into 2 or 4 ReflectIons* Ⅸ AutomorphIsms of the UnIt DIsc* Ⅹ EXercIses 4 DIfferentIatIon: The AmplItwIst Concept Ⅰ IntroductIon Ⅱ A PuzzlIng Phenomenon Ⅲ Local DescrIptIon of MappIngs In the Plane Ⅳ The CompleX Derivative as AmplItwIst Ⅴ Some SImple EXamples Ⅵ Conformal = AnalytIc Ⅶ CrItIcal PoInts Ⅷ The Cauchy-RIemann EquatIons Ⅸ EXercIses 5 Further Geometry of DIfferentIatIon Ⅰ Cauchy-RIemann ReVealed Ⅱ An IntImatIon of RIgIdIty Ⅲ Visual DIfferentIatIon of log(z) Ⅳ Rules of DIfferentIatIon Ⅴ PolynomIals, Power SerIes, and RatIonal Func-tIons Ⅵ Visual DIfferentIatIon of the Power FunctIon Ⅶ Visual DIfferentIatIon of eXp(z) 231 Ⅷ GeometrIc SolutIon of E'= E Ⅸ An ApplIcatIon of HIgher Derivatives: CurVa-ture* Ⅹ CelestIal MechanIcs* Ⅺ AnalytIc ContInuatIon* Ⅻ EXercIses 6 Non-EuclIdean Geometry* Ⅱ IntroductIon Ⅱ SpherIcal Geometry Ⅲ HyperbolIc Geometry Ⅳ EXercIses 7 WIndIng Numbers and Topology Ⅰ WIndIng Number Ⅱ Hopf's Degree Theorem Ⅲ PolynomIals and the Argument PrIncIple Ⅳ A TopologIcal Argument PrIncIple* Ⅴ Rouché's Theorem Ⅵ MaXIma and MInIma Ⅶ The Schwarz-PIck Lemma* Ⅷ The GeneralIzed Argument PrIncIple Ⅸ EXercIses 8 CompleX IntegratIon: Cauchy's Theorem ⅡntroductIon …… |
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