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 Phragmén-Lindelöf principle 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