Holomorphic function

Mathematical analysis → Complex analysis
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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 ⁠${\displaystyle \mathbb {C} ^{n}}$⁠. 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 ⁠${\displaystyle z_{0}}$⁠" means not just differentiable at ⁠${\displaystyle z_{0}}$⁠, but differentiable everywhere within some close neighbourhood of ⁠${\displaystyle z_{0}}$⁠ in the complex plane.