| 词条 | Antiholomorphic function |
| 释义 | {{Unreferenced|date=December 2009}} In mathematics, antiholomorphic functions (also called antianalytic functions) are a family of functions closely related to but distinct from holomorphic functions. A function of the complex variable z defined on an open set in the complex plane is said to be antiholomorphic if its derivative with respect to {{overline|z}} exists in the neighbourhood of each and every point in that set, where {{overline|z}} is the complex conjugate. One can show that if f(z) is a holomorphic function on an open set D, then f({{overline|z}}) is an antiholomorphic function on {{overline|D}}, where {{overline|D}} is the reflection against the x-axis of D, or in other words, {{overline|D}} is the set of complex conjugates of elements of D. Moreover, any antiholomorphic function can be obtained in this manner from a holomorphic function. This implies that a function is antiholomorphic if and only if it can be expanded in a power series in {{overline|z}} in a neighborhood of each point in its domain. If a function is both holomorphic and antiholomorphic, then it is constant on any connected component of its domain. {{DEFAULTSORT:Antiholomorphic Function}}{{mathanalysis-stub}} 2 : Complex analysis|Types of functions |
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