Complex analysis | Types of functions

Antiholomorphic function

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 z exists in the neighbourhood of each and every point in that set, where z is the complex conjugate. A definition of antiholomorphic function follows: "[a] function of one or more complex variables [is said to be anti-holomorphic if (and only if) it] is the complex conjugate of a holomorphic function ." One can show that if f(z) is a holomorphic function on an open set D, then f(z) is an antiholomorphic function on D, where D is the reflection against the x-axis of D, or in other words, 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 z in a neighborhood of each point in its domain. Also, a function f(z) is antiholomorphic on an open set D if and only if the function f(z) is holomorphic on D. If a function is both holomorphic and antiholomorphic, then it is constant on any connected component of its domain. (Wikipedia).

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Related pages

Complex conjugate | Complex plane | If and only if | Connected space | Mathematics | Function (mathematics) | Derivative | Power series | Open set | Holomorphic function