Holomorphic function

Mathematical analysis → Complex analysis
Complex analysis
Complex numbers
Real number Imaginary number Complex plane Complex conjugate Unit complex number
Complex functions
Complex-valued function Analytic function Holomorphic function Cauchy–Riemann equations Formal power series
Basic theory
Zeros and poles Cauchy's integral theorem Local primitive Cauchy's integral formula Winding number Laurent series Isolated singularity Residue theorem Argument principle Conformal map Schwarz lemma Harmonic function Laplace's equation
Geometric function theory
People
Augustin-Louis Cauchy Leonhard Euler Carl Friedrich Gauss Jacques Hadamard Kiyoshi Oka Bernhard Riemann Karl Weierstrass
Mathematics portal
v t e

In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space Cⁿ. The existence of a complex derivative in a neighbourhood is a very strong condition: it implies that a holomorphic function is infinitely differentiable and locally equal to its own Taylor series (is analytic). Holomorphic functions are the central objects of study in complex analysis.

Though the term analytic function is often used interchangeably with "holomorphic function", the word "analytic" is defined in a broader sense to denote any function (real, complex, or of more general type) that can be written as a convergent power series in a neighbourhood of each point in its domain. That all holomorphic functions are complex analytic functions, and vice versa, is a major theorem in complex analysis.^[1]

Holomorphic functions are also sometimes referred to as regular functions.^[2] A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic at a point z₀" means not just differentiable at z₀, but differentiable everywhere within some neighbourhood of z₀ in the complex plane.