Complex Analysis

Author:              Zhiming Ou

                         Mattermatics Learning Center

Publisher:       3265 Public Way

ISBN: 978-1-0677109-2-7

Summary

Complex analysis studies functions of complex variables. It provides means to prove the fundamental theorem of algebra; it solves difficult real integrals; it links prime numbers with the non-trivial zeros of the Riemann Zeta function. Most importantly, it expresses any types of functions freely! Real-world problems in physics and engineering (mechanic and genetic) in an elegant way. We can say that, complex analysis is the greatest invention of human.

In this book, I summarized all known results in the subject, and developed new skills to express functions, which lead to the proof of the Riemann Hypothesis, and the Lindelof conjecture; further, this leads to the proof of Goldbach conjecture.

I also chose some challenge problems from Putnam contests as examples to show how people create new functions.

 

Content

Chapter 1 Complex Functions, Limits and Continuity     3

  • 1.1 Point Sets in the Complex Plane 4
  • 1.2 Variables and Functions 6
  • 1.3 Limits of Functions 11

 

Chapter 2 Complex Differentiation       15

  • 2.1 Derivatives 15

Cauchy-Riemann Equations         16

  • 2.2 Differentials 17

 

Chapter 3 Complex Integration        21

  • 3.1 Definition 21

Cauchy’s Integral Theorem     23

  • 3.2 Indefinite Integrals 24
  • 3.3 Some Consequences of Cauchy's Theorem 25
  • 3.4 Cauchy's integral formula 27
  • 3.5 Two-Dimensional Vector Fields 30

 

Chapter 4 Infinite Series      32

  • 4.1 Concepts 32
  • 4.2 Power Series 34
  • 4.3 Laurent series 37
  • 4.4 Analytic Continuation 42
  • 4.5 Residues 44

 

Chapter 5 Infinite Product        49

  • 5.1 Definition 49
  • 5.2 Product Representations of Functions 50
  • 5.3 The Gamma Function 54
  • 5.4 Asymptotic Expansion 58

 

Chapter 6 Dirichlet Series       59

  • 6.1 General Dirichlet Series 59
  • 6.2 Riemann Zeta Function 62

 

Chapter 7 Conformal Mapping      67

  • 7.1 Real Transformations or Mappings 67
  • 7.2 Complex Mapping Functions 68