Author's Preface
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Preface by the editor

introduction

Chapter 1: Foundations of Complex Analysis
1 Complex numbers and the complex plane
1.1 Basic properties
1.2 Convergence
1.3 Sets in the complex plane
2 Functions in the complex plane
2.1 Continuous functions
2.2 Regular functions
2.3 Power series
3 Integral along a curve
Practice problems

Chapter 2 Cauchy's Theorem and Its Applications
1. Gursha's summary
2 Local existence of primitive functions on disks and Cauchy's theorem
3 Some integral calculations
4 Cauchy integral formula
5 Additional Applications
5.1 Morera's theorem
5.2 Regular function sequence
5.3 Complex analytic functions defined by integrals
5.4 Schwarz reflection principle
5.5 Runge approximation theorem
Practice problems
Advanced problems

Chapter 3: Glassy Functions and Logarithmic Functions
1 Zero and pole
2. Flow formula
2.1 Some examples
3 Singularities and Glassy Functions
4. Principle and Application of Declination
5 Continuous deformation and simple connection region
6 Complex logarithmic function
7 Fourier series and harmonic functions
Practice problems
Advanced problems

Chapter 4 Fourier Transform
1 Function set F
The action of the Fourier transform in 2 F
3 Paley-Wiener theorem
Practice problems
Advanced problems

Chapter 5: Pre-processing functions
1 Jensen formula
2 Functions with finite increasing exponents
3 infinite product
3.1 General Facts
3.2 Example: Product Formula for the Sine Function
4 Weierstrass infinite product
5 Hadamard Factorization Theorem
Practice problems
Advanced problems

Chapter 6 Gamma and Zeta Functions
1 Gamma function
1.1 Analytical continuity
1.2 Additional properties of Γ
2 Zeta function
2.1 Functional equations and analytical continuity
Practice problems
Advanced problems

Chapter 7: Zeta Function and Prime Number Theorem
1 Zero of the zeta function
1.1 Estimation of 1/ζ (s)
2 functions ψ and ψ1
2.1 Proof of the asymptotic equation for ψ1
3 Notes on double series
Practice problems
Advanced problems

Chapter 8 Conformal Mapping
1. Isomorphism and specific examples
1.1 The original plate and the upper half plane
1.2 Another example
1.3 Dirichlet problem on a belt
2 Schwarz's Lemma: Automorphism of a disk and its antiplane
2.1 Automorphism of the original plate
2.2 Automorphism of the antiplane
3 Riemann theorem
3.1 Description and necessary conditions of the Riemann mapping theorem
3.2 Montel's theorem
3.3 Proof of the Riemann mapping theorem
Conformal mapping to 4 polygons
4.1 Some examples
4.2 Schwarz-Christoffel integral
4.3 Behavior at the boundary
4.4 Expressions representing thoughts
4.5 Back to the elliptic integral
Practice problems
Advanced problems

Chapter 9: Introduction to Elliptic Functions
1 Elliptic function
1.1 Riouville Theorem
1.2 Weierstrass p function
2 Modularity of elliptic functions and Eisenstein series
2.1 Eisenstein series
2.2 Eisenstein series and divisor functions
Practice problems
Advanced problems

Chapter 10 Applications of the Theta Function
1 Product formula for Jacobi theta function
1.1 See more conversion rules
2 Generating function
Theorem on the sum of three square numbers
3.1 Two-square theorem
3.2 Four-square theorem
Practice problems
Advanced problems
Appendix A Asymptotic Estimation
1 Bessel function
2 Laplace's Method: Stirling's Formula
3 Airy function
4 partition function
Advanced problems

Appendix B Simple Connectivity and Jordan Curve Theorem
1 Propositions equivalent to simple connectivity
2 Jordan Curve Theorem
2.1 Proof of the general Cauchy theorem

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