• Source: 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




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

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