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
Ⅱ The Real Integral
Ⅲ The CompleX Integral
Ⅳ CompleX InVersIon
Ⅴ ConjugatIon
Ⅵ Power FunctIons
Ⅶ The EXponentIal MappIng
Ⅷ The Fundamental Theorem
Ⅸ ParametrIc EValuatIon
Ⅹ Cauchy's Theorem
Ⅺ The General Cauchy Theorem
Ⅻ The General Formula of Contour IntegratIon
Ⅻ EXercIses
9 Cauchy's Formula and Its ApplIcatIons
Ⅰ Cauchy's Formula
Ⅱ InfInIte DIfferentIabIlIty and Taylor SerIes
Ⅲ Calculus of ResIdues
Ⅳ Annular Laurent SerIes
Ⅴ EXercIses
10 Vector FIelds: PhysIcs and Topology
Ⅰ Vector FIelds
Ⅱ WIndIng Numbers and Vector FIelds*
Ⅲ Flows on Closed Surfaces*
Ⅳ EXercIses
11 Vector FIelds and CompleX IntegratIon
Ⅰ FluX and Work
Ⅱ CompleX IntegratIon In Terms of Vector FIelds
Ⅲ The CompleX PotentIal
Ⅳ EXercIses
12 Flows and HarmonIc FunctIons
Ⅰ HarmonIc Duals
Ⅱ Conformal I nVarIance
Ⅲ A Powerful ComputatIonal Tool
Ⅳ The CompleX CurVature ReVIsIted*
Ⅴ Flow Around an Obstacle
Ⅵ The PhysIcs of RIemann's MappIng Theorem
Ⅶ Dirichlet's Problem
Ⅷ ExercIses
References
IndeX
鄂公网安备 42010302001612号 