Holomorphic Function

Holomorphic Function

In mathematics, holomorphic functions are the central objects of study in complex analysis. A holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighborhood of every point in its domain. The existence of a complex derivative is a very strong condition, for it implies that any holomorphic function is actually infinitely differentiable and equal to its own Taylor series.

The term analytic function is often used interchangeably with “holomorphic function”, although the word “analytic” is also used in a broader sense to describe any function (real, complex, or of more general type) that is equal to its Taylor series in a neighborhood of each point in its domain. The fact that the class of complex analytic functions coincides with the class of holomorphic functions is a major theorem in complex analysis.

Holomorphic functions are also sometimes referred to as regular functions or as conformal maps. 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 neighborhood of z₀ in the complex plane.