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アルフォースの『複素解析』を買う。

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ラース・ヴァレリアン・アールフォルス - Wikipedia


Lars V. Ahlfors
『COMPLEX ANALYSIS』

Contents

chapter 1 complex numbers
1 the algebra of complex numbers
1.1 arithmetic operations
1.2 square roots
1.3 justification
1.4 conjugation, absolute value
1.5 inequalities
2 the geometric representation of complex numbers
2.1 geometric addition and multiplication
2.2 the binomial equation
2.3 analytic geomety
2.4 the spherical representation

chapter 2 complex functions
1 introduction to the concept of analytic function
1.1 limits and continuity
1.2 analytic functions
1.3 polynomials
1.4 rational functions
2 elementary theory of power series
2.1 sequences
2.2 series
2.3 uniform convergence
2.4 power series
2.5 abel's limit theorem
3 the exponential and trigonometric functions
3.1 the exponential
3.2 the trigonometric functions
3.3 the periodicity
3.4 the logarithm

chapter 3 analytic functions as mapping
1 elementary point set topology
1.1 sets and elements
1.2 metric spaces
1.3 connectedness
1.4 compactness
1.5 continuous functions
1.6 topological spsces
2 conformality
2.1 arcs and closed curves
2.2 analytic functions in regions
2.3 conformal mapping
2.4 length and area
3 linear transformations
3.1 the linear group
3.2 the cross ratio
3.3 symmetry
3.4 oriented circles
3.5 families of circles
4 elementary conformal mappings
4.1 the use of level curves
4.2 a survey of elementary mappings
4.3 elementary reimann surfaces

chapter 4 complex integration
1 fundamental theorems
1.1 line integrals
1.2 rectifiable arcs
1.3 line integrals ad functions of arcs
1.4 cauchy's theorem for a rectangle
1.5 cauchy's theorem in a disk
2 cauchy's integral formula
2.1 the index of a point with respect to a closed curve
2.2 the integral formula
2.3 higher derivatives
3 local properties of analytical functions
3.1 removable singularities. taylor's theorem
3.2 zeros and poles
3.3 the local mapping
3.4 the maximum principle
4 the general form of cauchy's theorem
4.1 chains and cycles
4.2 simple connectivity
4.3 homology
4.4 the general statement of cauchy's theorem
4.5 proof of cauchy's theorem
4.6 locally exact differentials
4.7 multiply connected regions
5 the caluculus of residues
5.1 the residue theorem
5.2 the argument principle
5.3 evaluation of definite integrals
6 harmonic functions
6.1 definition and basic properties
6.2 the mean-value property
6.3 poisson's formula
6.4 schwarz's theorem
6.5 the reflection principle

chapter 5 series and product drvelopments
1 power series expansions
1.1 weierstrass's theorem
1.2 the taylor series
1.3 the laurent series
2 partial fractions and factorization
2.1 partial fractions
2.2 infinite products
2.3 canonical products
2.4 the gamma function
2.5 stirling's formula
3 entire functions
3.1 jensen's formula
3.2 hadamard's theorem
4 the reimann zeta function
4.1 the product development
4.2 extension of ζ(s) to the whole plane
4.3 the functional equation
4.4 the zeros of the zeta function
5 normal families
5.1 equicontinuity
5.2 normality and compactness
5.3 arzela's theorem
5.4 families of analytic functions
5.5 the classical definition

chapter 6 conformal mapping. dirichlet's problem
1 the families mapping theorem
1.1 statement and proof
1.2 boundary behavior
1.3 use of the reflection principle
1.4 analytic arcs
2 conformal mapping of polygons
2.1 the behavior at an angle
2.2 the schwarz-christoffel formula
2.3 mapping on a rectangle
2.4 the triagle functions of schwarz
3 a closer look at harmonic functions
3.1 functions with the mean-value property
3.2 harnack's principle
4 the dirichlet problem
4.1 subharmonic functions
4.2 solution of dirichlet's problem
5 canonical mapping of maltiply connected regions
5.1 harmonic measures
5.2 green's function
5.3 parallel slit regions

chapter 7 elliptic functions
1 simply periodic functions
1.1 representation by exponentials
1.2 the fourier development
1.3 functions of finite order
2 doubly periodic functions
2.1 the priod module
2.2 unimodular transformations
2.3 than canonical basis
2.4 general properties of elliptic functions
3 the weierstrass theory
3.1 the weierstrass ρ-function
3.2 the functions ζ(z) and σ(z)
3.3 differential equation
3.4 the modular function λ(τ)
3.5 the conformal mapping λ(τ)

chapter 8 global analytic functions
1 analytic continuation
1.1 the weierstrass theory
1.2 germs and sheaves
1.3 sections and reimann surfaces
1.4 analytic continuations along arcs
1.5 homotopic curves
1.6 the monodoromy theorem
1.7 branch points
2 algebraic functions
2.1 the resultant of two polynomials
2.2 definition and properties of algebraic functions
2.3 behavior at the critical points
3 picard's theorem
3.1 lacunary values
4 linear differential equation
4.1 ordinary points
4.2 regular singular points
4.3 solutions at infinity
4.4 the hypergeometric differential equqtion
4.5 reimann's point of view


Complex Analysis

Complex Analysis