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Main Question or Discussion Point
- Author: Elias Stein, Rami Shakarchi
- Title: Complex Analysis
- Amazon Link: https://www.amazon.com/dp/0691113858/?tag=pfamazon01-20
- Prerequisities: Fourier Analysis by Stein, Shakarchi
- Level: Undergrad
Table of Contents:
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[LIST]
[*] Foreword
[*] Introduction
[*] Preliminaries to Complex Analysis
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[*] Complex numbers and the complex plane
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[*] Basic properties
[*] Convergence
[*] Sets in the complex plane
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[*] Functions on the complex plane
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[*] Continuous functions
[*] Holomorphic functions
[*] Power series
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[*] Integration along curves
[*] Exercises
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[*] Cauchy's Theorem and Its Applications
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[*] Goursat's theorem
[*] Local existence of primitives and Cauchy's theorem in a disc
[*] Evaluation of some integrals
[*] Cauchy's integral formulas
[*] Further applications
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[*] Morera's theorem
[*] Sequences of holomorphic functions
[*] Holomorphic functions defined in terms of integrals
[*] Schwarz reflection principle
[*] Runge's approximation theorem
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[*] Exercises
[*] Problems
[/LIST]
[*] Meromorphic Function and the Logarithm
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[*] Zeros and Poles
[*] The residue formula
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[*] Examples
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[*] Singularities and meromorphic functions
[*] The argument principle and applications
[*] Homotopies and simply connected domains
[*] The complex logarithm
[*] Fourier series and harmonic functions
[*] Exercises
[*] Problems
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[*] The Fourier Transform
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[*] The class F
[*] Action of the Fourier transform on F
[*] Paley-Wiener theorem
[*] Exercises
[*] Problems
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[*] Entire Functions
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[*] Jensen's formula
[*] Functions of finite order
[*] Infinite products
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[*] Generalities
[*] Example: the product formula for the sine function
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[*] Weierstrass infinite products
[*] Hadamard's factorization theorem
[*] Exercises
[*] Problems
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[*] The Gamma and Zeta Functions
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[*] The gamma function
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[*] Analytic continuation
[*] Further properties of \Gamma
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[*] The zeta function
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[*] Functional equation and analytic continuation
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[*] Exercises
[*] Problems
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[*] The Zeta Function and Prime Number Theorem
[LIST]
[*] Zeros of the zeta function
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[*] Estimates for 1/\zeta(s)
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[*] Reduction to the functions \psi and \psi_1
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[*] Proofs of the asymptotics for \psi_1
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[*] Note on interchanging double sums
[*] Exercises
[*] Problems
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[*] Conformal Mappins
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[*] Conformal equivalence and examples
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[*] The disc and upper half-plane
[*] Further examples
[*] The Dirichlet problem in a strip
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[*] The Schwarz lemma; automorphisms of the disc and upper half-plane
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[*] Automorphisms of the disc
[*] Automorphisms of the upper half-plane
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[*] The Riemann mapping theorem
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[*] Necessary conditions and statement of the theorem
[*] Montel's theorem
[*] Proof of the Riemann mapping theorem
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[*] Conformal mappings onto polygons
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[*] Some examples
[*] The Schwarz-Christoffel integral
[*] Boundary behavior
[*] The mapping formula
[*] Return to elliptic integrals
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[*] Exercises
[*] Problems
[/LIST]
[*] An Introduction to Elliptic Functions
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[*] Elliptic functions
[LIST]
[*] Liouville's theorems
[*] The Weierstrass P function
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[*] The modular character of elliptic functions and Eisenstein series
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[*] Eisenstein series
[*] Eisenstein series and divisor functions
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[*] Exercises
[*] Problems
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[*] Applications of Theta Functions
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[*] Product formula for the Jacobi theta function
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[*] Further transformation laws
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[*] Generating functions
[*] The theorems about sums of squares
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[*] The two-squares theorem
[*] The four-squares theorem
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[*] Exercises
[*] Problems
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[*] Appendix: Asymptotics
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[*] Bessel functions
[*] Laplace method; Stirling's formula
[*] The Airy function
[*] The partition function
[*] Problems
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[*] Simple Connectivity and Jordan Curve Theorem
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[*] Equivalent formulations of simple connectivity
[*] The Jordan curve theorem
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[*] Proof of a general form of Cauchy's theorem
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[*] Notes and References
[*] Bilbiography
[*] Symbol Glossary
[*] Index
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