Learn how and when to remove this template message 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. 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.

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As a consequence of the Cauchy—Riemann equations , a real-valued holomorphic function must be constant. Therefore, the absolute value of z, the argument of z, the real part of z and the imaginary part of z are not holomorphic.

Another typical example of a continuous function which is not holomorphic is the complex conjugate z formed by complex conjugation. Several variables[ edit ] The definition of a holomorphic function generalizes to several complex variables in a straightforward way. The function f is analytic at a point p in D if there exists an open neighbourhood of p in which f is equal to a convergent power series in n complex variables.

More generally, a function of several complex variables that is square integrable over every compact subset of its domain is analytic if and only if it satisfies the Cauchy—Riemann equations in the sense of distributions. Functions of several complex variables are in some basic ways more complicated than functions of a single complex variable. For example, the region of convergence of a power series is not necessarily an open ball; these regions are Reinhardt domains , the simplest example of which is a polydisk.

However, they also come with some fundamental restrictions. Unlike functions of a single complex variable, the possible domains on which there are holomorphic functions that cannot be extended to larger domains are highly limited. Such a set is called a domain of holomorphy.

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## Antiholomorphic function

As a consequence of the Cauchy—Riemann equations , a real-valued holomorphic function must be constant. Therefore, the absolute value of z, the argument of z, the real part of z and the imaginary part of z are not holomorphic. Another typical example of a continuous function which is not holomorphic is the complex conjugate z formed by complex conjugation. Several variables[ edit ] The definition of a holomorphic function generalizes to several complex variables in a straightforward way.

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## Fonction holomorphe

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## Fonctions d'une variable complexe/Fonctions holomorphes

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