In mathematics, antiholomorphic functions (also called antianalytic functionsEncyclopedia of Mathematics, Springer and The European Mathematical Society, https://encyclopediaofmath.org/wiki/Anti-holomorphic_function, As of 11 September 2020, This article was adapted from an original article by E. D. Solomentsev (originator), which appeared in Encyclopedia of Mathematics, .) are a family of functions closely related to but distinct from

holomorphic function In mathematics, a holomorphic function is a complex-valued function of one or more complex variables that is complex differentiable in a neighbourhood of each point in a domain in complex coordinate space . The existence of a complex de ...

s. A function of the complex variable z defined on an

open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that a ...

in the

complex plane In mathematics, the complex plane is the plane formed by the complex numbers, with a Cartesian coordinate system such that the -axis, called the real axis, is formed by the real numbers, and the -axis, called the imaginary axis, is formed by th ...

is said to be antiholomorphic if its

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...

with respect to exists in the neighbourhood of each and every point in that set, where is the

complex conjugate In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - ...

. A definition of antiholomorphic function follows:

" function $f (z) = u + i v$ of one or more complex variables $z = \left(z_1, \dots, z_n\right) \in \Complex^n$
s said to be anti-holomorphic if (and only if) it S, or s, is the nineteenth letter in the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''ess'' (pronounced ), plural ''esses''. Histo ...
is the complex conjugate of a holomorphic function $\overline = u - i v$ ."

One can show that if ''f''(''z'') is a

on an open set ''D'', then ''f''() is an antiholomorphic function on , where is the reflection against the ''x''-axis of ''D'', or in other words, 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 In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false. The connective is bi ...

it can be expanded in a

power series In mathematics, a power series (in one variable) is an infinite series of the form \sum_^\infty a_n \left(x - c\right)^n = a_0 + a_1 (x - c) + a_2 (x - c)^2 + \dots where ''an'' represents the coefficient of the ''n''th term and ''c'' is a con ...

in 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.

References

Complex analysis Types of functions {{mathanalysis-stub