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.

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

History

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