Holomorphic function

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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[1] 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 neighbourhood of z0 in the complex plane.

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Definition

Given a complex-valued function ƒ of a single complex variable, the derivative of ƒ at a point z0 in its domain is defined by the limit

This is the same as the definition of the derivative for real functions, except that all of the quantities are complex. In particular, the limit is taken as the complex number z approaches z0, and must have the same value for any sequence of complex values for z that approach z0 on the complex plane. If the limit exists, we say that ƒ is differentiable at the point z0. This concept of complex differentiability shares several properties with real differentiability: it is linear and obeys the product rule, quotient rule, and chain rule.