# List of complex analysis topics

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematics that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, and electrical engineering.
See also: glossary of real and complex analysis.

## Overview

* Complex numbers
* Complex plane
* Complex functions
** Complex derivative
** Holomorphic functions
** Harmonic functions
* Elementary functions
** Polynomial functions
** Exponential functions
** Trigonometric functions
** Hyperbolic functions
** Logarithmic functions
** Inverse trigonometric functions
** Inverse hyperbolic functions
* Residue theory
* Isometries in the complex plane

## Related fields

* Number theory
* Hydrodynamics
* Thermodynamics
* Electrical engineering

## Local theory

* Holomorphic function
* Antiholomorphic function
* Cauchy–Riemann equations
* Conformal mapping
** Conformal welding
* Power series
* Radius of convergence
* Laurent series
* Meromorphic function
* Entire function
* Pole (complex analysis)
* Zero (complex analysis)
* Residue (complex analysis)
* Isolated singularity
* Removable singularity
* Essential singularity
* Branch point
* Principal branch
* Weierstrass–Casorati theorem
* Landau's constants
* Holomorphic functions are analytic
* Schwarzian derivative
* Analytic capacity
* Disk algebra
* Univalent function

## Growth and distribution of values

* Ahlfors theory
* Bieberbach conjecture
* Borel–Carathéodory theorem
* Corona theorem
* Hadamard three-circle theorem
* Hardy space
* Hardy's theorem
* Maximum modulus principle
* Nevanlinna theory
* Paley–Wiener theorem
* Progressive function
* Value distribution theory of holomorphic functions

## Contour integrals

* Line integral
* Cauchy's integral theorem
* Cauchy's integral formula
* Residue theorem
* Liouville's theorem (complex analysis)
* Examples of contour integration
* Fundamental theorem of algebra
* Simply connected
* Winding number
** Principle of the argument
** Rouché's theorem
* Bromwich integral
* Morera's theorem
* Mellin transform
* Kramers–Kronig relation, a. k. a. Hilbert transform
* Sokhotski–Plemelj theorem

## Special functions

* Exponential function
* Beta function
* Gamma function
* Riemann zeta function
** Riemann hypothesis
** Generalized Riemann hypothesis
* Elliptic function
** Half-period ratio
** Jacobi's elliptic functions
** Weierstrass's elliptic functions
** Theta function
* Elliptic modular function
* J-function
* Modular function
* Modular form

## Riemann surfaces

* Analytic continuation
* Riemann sphere
* Riemann surface
* Riemann mapping theorem
* Carathéodory's theorem (conformal mapping)
* Riemann–Roch theorem

## Other

* Amplitwist
* Antiderivative (complex analysis)
* Bôcher's theorem
* Cayley transform
* Harmonic conjugate
* Hilbert's inequality
* Method of steepest descent
* Montel's theorem
* Periodic points of complex quadratic mappings
* Pick matrix
* Runge approximation theorem
* Schwarz lemma
* Weierstrass factorization theorem
* Mittag-Leffler's theorem
* Sendov's conjecture
* Infinite compositions of analytic functions

## Several complex variables

* Biholomorphy
* Cartan's theorems A and B
* Cousin problems
* Edge-of-the-wedge theorem
* Several complex variables

## People

* Augustin Louis Cauchy
* Leonhard Euler
* Carl Friedrich Gauss
* Jacques Hadamard
* Kiyoshi Oka
* Bernhard Riemann
* Karl Weierstrass
* Pierre Alphonse Laurent
* Brook Taylor
* Siméon Denis Poisson
* Hermann Schwarz
* Camille Jordan
* Carl Gustav Jacob Jacobi
* Eugène Rouché
* Gerardus Mercator
* Joseph Liouville
* Pierre-Simon Laplace
* August Ferdinand Möbius
* William Kingdon Clifford