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 \bar z exists in the neighbourhood of each and every point in that set, where \bar z is the complex conjugate of z . A definition of antiholomorphic function follows: "[a] function f(z) = u + i v of one or more complex variables z = \left(z_1, \dots, z_n\right) \in \Complex^n [is said to be anti-holomorphic if (and only if) it] is the complex conjugate of a holomorphic function \overline{f \left(z\right)} = u - i v ." One can show that if f(z) is a holomorphic function on an open set D , then f(\bar z) is an antiholomorphic function on \bar D , where \bar D is the reflection of D across the real axis; in other words, \bar 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 \bar 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 \overline{f(z)} is holomorphic on D .…

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 \bar z exists in the neighbourhood of each and every point in that set, where \bar z is the complex conjugate of z . A definition of antiholomorphic function follows: "[a] function f(z) = u + i v of one or more complex variables z = \left(z_1, \dots, z_n\right) \in \Complex^n [is said to be anti-holomorphic if (and only if) it] is the complex conjugate of a holomorphic function \overline{f \left(z\right)} = u - i v ." One can show that if f(z) is a holomorphic function on an open set D , then f(\bar z) is an antiholomorphic function on \bar D , where \bar D is the reflection of D across the real axis; in other words, \bar 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 \bar 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 \overline{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. ==References==

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