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Concept
Holomorphic function
Summary
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 (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.
Holomorphic functions are also sometimes referred to as regular functions. 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.
Given a complex-valued function f of a single complex variable, the derivative of f at a point z0 in its domain is defined as the limit
This is the same definition as for the derivative of a real function, except that all quantities are complex. In particular, the limit is taken as the complex number z tends to z0, and this means that the same value is obtained for any sequence of complex values for z that tends to z0. If the limit exists, f is said to be complex differentiable at 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.
A function is holomorphic on an open set U if it is complex differentiable at every point of U. A function f is holomorphic at a point z0 if it is holomorphic on some neighbourhood of z0.
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