Complex Analysis

Theory of Functions of a Complex Variable

What is Complex Analysis?

Complex analysis (also known as the theory of functions of a complex variable) is a branch of mathematics that studies functions whose variables are complex numbers. It reveals strikingly beautiful properties that have no counterpart in real analysis, and many results are celebrated as among the most elegant in all of mathematics.

Figure 1: The Complex Plane and Complex Functions

Complex Plane (z-plane) Re Im z = x + iy f(z) Image Plane (w-plane) Re Im w = f(z)

Differences from Real Analysis

The condition that a complex function is "differentiable" (holomorphic) is far stronger than in the real case. Holomorphic functions are automatically infinitely differentiable and can be expanded as Taylor series. From this property, the beautiful theorems unique to complex analysis follow.

  • Cauchy's Integral Theorem: The integral of a holomorphic function over a closed curve is zero
  • Cauchy's Integral Formula: Function values are determined by integrals over the boundary
  • Liouville's Theorem: Every bounded entire function is constant
  • Residue Theorem: Enables easy computation of complex integrals

Figure 2: Cauchy's Integral Theorem

D (holomorphic region) C $\displaystyle\oint_C f(z)\,dz = 0$

Where is Complex Analysis Used?

  • Physics: Fluid dynamics, electromagnetism, quantum mechanics
  • Engineering: Signal processing, control theory, electrical circuits
  • Mathematics: Number theory (zeta function), algebraic geometry, differential equations
  • Applications: Solving boundary value problems via conformal mappings

This series covers complex analysis systematically in four stages: from complex number basics through holomorphic functions, Cauchy's theorems, residue calculus, and conformal mappings.

Contents by Level

Introductory

Complex Numbers and Complex Functions

  • Definition and arithmetic of complex numbers
  • The complex plane and polar form
  • Euler's formula
  • Basics of complex functions
  • Complex exponential and trigonometric functions
  • Complex logarithm
High school – 1st year university

Elementary

Holomorphic Functions and Cauchy's Theorems

  • Complex differentiation and holomorphic functions
  • Cauchy–Riemann equations
  • Contour integration
  • Cauchy's integral theorem
  • Cauchy's integral formula
  • Power series and analytic functions
1st – 2nd year university

Intermediate

Singularities and Residues

  • Laurent series
  • Classification of singularities
  • Residue computation
  • Residue theorem
  • Applications to real integrals
  • Argument principle
2nd – 3rd year university

Advanced

Conformal Mappings and Advanced Topics

  • Theory of conformal mappings
  • Riemann mapping theorem
  • Schwarz–Christoffel transformation
  • Analytic continuation
  • Gamma function and zeta function
  • Introduction to elliptic functions
3rd year university – graduate

Key Concepts and Formulas

Euler's Formula

$$e^{i\theta} = \cos\theta + i\sin\theta$$

Cauchy–Riemann Equations

$$\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}, \quad \frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}$$

Cauchy's Integral Formula

$$f(a) = \frac{1}{2\pi i}\oint_C \frac{f(z)}{z-a}\,dz$$

Residue Theorem

$$\oint_C f(z)\,dz = 2\pi i \sum_{k} \mathrm{Res}(f, z_k)$$

Laurent Series

$$f(z) = \sum_{n=-\infty}^{\infty} a_n (z-a)^n$$

Argument Principle

$$\frac{1}{2\pi i}\oint_C \frac{f'(z)}{f(z)}\,dz = N - P$$

($N$: number of zeros, $P$: number of poles)

Prerequisites

  • Introductory: High school mathematics (trigonometry, exponentials and logarithms)
  • Elementary: Introductory content, calculus (multivariable partial derivatives, line integrals)
  • Intermediate: Elementary content, convergence of series
  • Advanced: Intermediate content, basics of topological spaces

References