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 a Taylor series around each point. 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 otherwise difficult integrals—including real 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
- and more (8 chapters in all, excluding exercises)
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
- and more (14 chapters in all, excluding exercises)
Key Concepts and Formulas
Euler's Formula
$$e^{i\theta} = \cos\theta + i\sin\theta$$
Cauchy–Riemann Equations
$$\dfrac{\partial u}{\partial x} = \dfrac{\partial v}{\partial y}, \quad \dfrac{\partial u}{\partial y} = -\dfrac{\partial v}{\partial x}$$
(These hold when $f = u + iv$ is holomorphic; conversely, if the partial derivatives of $u, v$ are continuous and satisfy them, $f$ is holomorphic)
Cauchy's Integral Formula
$$f(a) = \dfrac{1}{2\pi i}\oint_C \dfrac{f(z)}{z-a}\,dz$$
($C$ a positively oriented simple closed curve; $f$ holomorphic on an open set containing $C$ and its interior; $a$ lies inside $C$)
Residue Theorem
$$\oint_C f(z)\,dz = 2\pi i \sum_{k} \mathrm{Res}(f, z_k)$$
($C$ a positively oriented simple closed curve; $f$ holomorphic on $C$, and inside $C$ except at finitely many isolated singularities $z_k$)
Laurent Series
$$f(z) = \sum_{n=-\infty}^{\infty} a_n (z-a)^n$$
(for $f$ holomorphic on an annulus $r \lt |z-a| \lt R$; the series holds on that annulus)
Argument Principle
$$\dfrac{1}{2\pi i}\oint_C \dfrac{f'(z)}{f(z)}\,dz = Z - P$$
($C$ a positively oriented simple closed curve; $f$ meromorphic on an open set containing $C$ and its interior, with no zero or pole on $C$; $Z$ and $P$ count the zeros and poles inside $C$, each with multiplicity)
Prerequisites
- Introductory: High school mathematics (trigonometry, exponentials and logarithms), basic calculus (the concept of limits)
- Elementary: Introductory content, calculus (multivariable partial derivatives, line integrals), basics of sequences, series and radius of convergence
- Intermediate: Elementary content, convergence of series (uniform and absolute convergence), basics of Taylor expansion
- Advanced: Intermediate content, basics of topological spaces (open sets, connectedness, compactness), basics of real analysis (uniform convergence, continuous functions on compact sets)
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 a Taylor series around each point.
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) and basic calculus (the concept of limits) are needed. Elementary level requires multivariable partial derivatives and line integrals, and the basics of sequences, series and radius of convergence. Intermediate level requires convergence of series (uniform and absolute convergence) and the basics of Taylor expansion. Advanced level requires basics of topological spaces and of real analysis (uniform convergence, continuous functions on compact sets).