複分析

《複分析：可視化方法(英文版)》是複分析領域近年來較有影響的一本著作。作者用豐富的圖例展示各種概念、定理和證明思路，十分便於讀者理解，充分揭示了複分析的數學之美。書中講述的內容有幾何、複變函數變換、默比烏斯變換、微分、非歐幾何、復積分、柯西公式、向量場、復積分、調和函數等。 ^[2] 

1 Geometry and CompleX ArIthmetIc

1 IntroductIon

2 Eulers Formula

3 Some ApplIcatIons

4 TransformatIons and EuclIdean Geometry*

5 EXercIses

2 CompleX FunctIons as TransformatIons

1 IntroductIon

2 PolynomIals

3 Power SerIes

4 The EXponentIal FunctIon

5 CosIne and SIne

6 MultIfunctIons

7 The LogarIthm FunctIon

8 AVeragIng oVer CIrcles*

8 EXercIses

3 M?bIus TransformatIons and InVersIon

1 IntroductIon

2 InVersIon

3 Three Illustrative ApplIcatIons of InVersIon

4 The RIemann Sphere

5 M?bIus TransformatIons： BasIc Results

6 M?bIus TransformatIons as MatrIces*

7 VisualIzatIon and ClassIfIcatIon*

8 DecomposItIon Into 2 or 4 ReflectIons*

8 AutomorphIsms of the UnIt DIsc*

9 EXercIses

4 DIfferentIatIon： The AmplItwIst Concept

1 IntroductIon

2 A PuzzlIng Phenomenon

3 Local DescrIptIon of MappIngs In the Plane

4 The CompleX Derivative as AmplItwIst

5 Some SImple EXamples

6 Conformal = AnalytIc

7 CrItIcal PoInts

8 The Cauchy-RIemann EquatIons

8 EXercIses

5 Further Geometry of DIfferentIatIon

1 Cauchy-RIemann ReVealed

2 An IntImatIon of RIgIdIty

3 Visual DIfferentIatIon of log(z)

4 Rules of DIfferentIatIon

5 PolynomIals， Power SerIes， and RatIonal Func-tIons

6 Visual DIfferentIatIon of the Power FunctIon

7 Visual DIfferentIatIon of eXp(z) 231

8 GeometrIc SolutIon of E= E

8 An ApplIcatIon of HIgher Derivatives： CurVa-ture*

9 CelestIal MechanIcs*

10 AnalytIc ContInuatIon*

11 EXercIses

6 Non-EuclIdean Geometry*

2 IntroductIon

2 SpherIcal Geometry

3 HyperbolIc Geometry

4 EXercIses

7 WIndIng Numbers and Topology

1 WIndIng Number

2 Hopfs Degree Theorem

3 PolynomIals and the Argument PrIncIple

4 A TopologIcal Argument PrIncIple*

5 Rouchés Theorem

6 MaXIma and MInIma

7 The Schwarz-PIck Lemma*

8 The GeneralIzed Argument PrIncIple

8 EXercIses

8 CompleX IntegratIon： Cauchys Theorem

2ntroductIon

2 The Real Integral

3 The CompleX Integral

4 CompleX InVersIon

5 ConjugatIon

6 Power FunctIons

7 The EXponentIal MappIng

8 The Fundamental Theorem

8 ParametrIc EValuatIon

9 Cauchys Theorem

10 The General Cauchy Theorem

11 The General Formula of Contour IntegratIon

11 EXercIses

9 Cauchys Formula and Its ApplIcatIons

1 Cauchys Formula

2 InfInIte DIfferentIabIlIty and Taylor SerIes

3 Calculus of ResIdues

4 Annular Laurent SerIes

5 EXercIses

10 Vector FIelds： PhysIcs and Topology

1 Vector FIelds

2 WIndIng Numbers and Vector FIelds*

3 Flows on Closed Surfaces*

4 EXercIses

11 Vector FIelds and CompleX IntegratIon

1 FluX and Work

2 CompleX IntegratIon In Terms of Vector FIelds

3 The CompleX PotentIal

4 EXercIses

12 Flows and HarmonIc FunctIons

1 HarmonIc Duals

2 Conformal I nVarIance

3 A Powerful ComputatIonal Tool

4 The CompleX CurVature ReVIsIted*

5 Flow Around an Obstacle

6 The PhysIcs of RIemanns MappIng Theorem

7 Dirichlets Problem

8 ExercIses

References

IndeX ^[2]