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 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(\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 is holomorphic on D. If a function is both holomorphic and antiholomorphic, then it is constant on any connected component of its domain.