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 far stronger than their real-variable counterparts, and many results are celebrated as among the most elegant in all of mathematics.
Figure 1: The Complex Plane and Complex Functions
Differences from Real Analysis
The condition that a complex function is complex-differentiable at every point of a domain — that is, holomorphic — is far stronger than differentiability 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 function holomorphic on a simply connected domain, taken over a closed curve in that domain, 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
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
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
- Maximum principle and Liouville's theorem
Intermediate
Singularities and Residues
- Laurent series
- Classification of singularities
- Residue computation
- Residue theorem
- Applications to real integrals
- Argument principle
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
Key Concepts and Formulas
Euler's Formula
$$e^{i\theta} = \cos\theta + i\sin\theta$$
Cauchy–Riemann Equations
$$\dfrac{\partial u}{\partial x} = \dfrac{\partial v}{\partial y}, \quad \dfrac{\partial u}{\partial y} = -\dfrac{\partial v}{\partial x}$$
Cauchy's Integral Formula
$$f(a) = \dfrac{1}{2\pi i}\oint_C \dfrac{f(z)}{z-a}\,dz$$
($f$ holomorphic on $C$ and its interior; $a$ lies inside $C$)
Residue Theorem
$$\oint_C f(z)\,dz = 2\pi i \sum_{k} \mathrm{Res}(f, z_k)$$
($z_k$ are the isolated singularities of $f$ inside $C$)
Laurent Series
$$f(z) = \sum_{n=-\infty}^{\infty} a_n (z-a)^n$$
Argument Principle
$$\dfrac{1}{2\pi i}\oint_C \dfrac{f'(z)}{f(z)}\,dz = N - P$$
($N$ and $P$ count the zeros and poles inside $C$, each with multiplicity; $f$ has no zero or pole on $C$)
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
Frequently Asked Questions
What is complex analysis?
Complex analysis (theory of functions of a complex variable) is a branch of mathematics that studies functions of complex variables. It reveals beautiful properties not seen in real analysis — for example, holomorphic functions are automatically infinitely differentiable and can be expressed as Taylor series.
Where is complex analysis used?
Complex analysis is used in physics (fluid dynamics, electromagnetism, quantum mechanics), engineering (signal processing, control theory, electrical circuits), mathematics (number theory, algebraic geometry, differential equations), and applied mathematics (solving boundary value problems via conformal mappings).
What prerequisites are needed to study complex analysis?
At the introductory level, high school mathematics (trigonometry, exponentials and logarithms) is needed. Elementary level requires multivariable partial derivatives and line integrals. Intermediate level requires convergence of series. Advanced level requires basics of topological spaces.