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 . 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. A holomorphic function whose domain is the whole complex plane is called an entire function. The phrase "holomorphic at a point " means not just differentiable at, but differentiable everywhere within some close neighbourhood of in the complex plane.
Given a complex-valued function of a single complex variable, the derivative of at a point in its domain is defined as the limit[2]
f'(z0)=
| \lim | |
| z\toz0 |
| f(z)-f(z0) | |
| z-z0 |
.
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 tends to, and this means that the same value is obtained for any sequence of complex values for that tends to . If the limit exists, is said to be complex differentiable at . This concept of complex differentiability shares several properties with real differentiability: It is linear and obeys the product rule, quotient rule, and chain rule.[3]
A function is holomorphic on an open set if it is complex differentiable at every point of . A function is holomorphic at a point if it is holomorphic on some neighbourhood of .[4] A function is holomorphic on some non-open set if it is holomorphic at every point of .
A function may be complex differentiable at a point but not holomorphic at this point. For example, the function
stylef(z)=|z|\vphantom{l}2=z\bar{z}
The relationship between real differentiability and complex differentiability is the following: If a complex function is holomorphic, then and have first partial derivatives with respect to and, and satisfy the Cauchy–Riemann equations:[5]
| \partialu | |
| \partialx |
=
| \partialv | |
| \partialy |
and
| \partialu | |
| \partialy |
=-
| \partialv | |
| \partialx |
| \partialf | |
| \partial\bar{z |
If continuity is not given, the converse is not necessarily true. A simple converse is that if and have continuous first partial derivatives and satisfy the Cauchy–Riemann equations, then is holomorphic. A more satisfying converse, which is much harder to prove, is the Looman–Menchoff theorem: if is continuous, and have first partial derivatives (but not necessarily continuous), and they satisfy the Cauchy–Riemann equations, then is holomorphic.[7]
The term holomorphic was introduced in 1875 by Charles Briot and Jean-Claude Bouquet, two of Augustin-Louis Cauchy's students, and derives from the Greek ὅλος (hólos) meaning "whole", and μορφή (morphḗ) meaning "form" or "appearance" or "type", in contrast to the term meromorphic derived from μέρος (méros) meaning "part". A holomorphic function resembles an entire function ("whole") in a domain of the complex plane while a meromorphic function (defined to mean holomorphic except at certain isolated poles), resembles a rational fraction ("part") of entire functions in a domain of the complex plane.