A complex functionf:D,  where D is a domain of the complex planeMathworldPlanetmath, having the derivativeMathworldPlanetmath


in each point z of D, is said to be antiholomorphic in D.

The following conditions are equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmath (

  • f(z) is antiholomorphic in D.

  • f(z)¯  is holomorphic in D.

  • f(z¯) is holomorphic in  D¯:={z¯zD}.

  • f(z) may be to a power seriesMathworldPlanetmath n=0an(z¯-u)n at each  uD.

  • The real partu(x,y)  and the imaginary partv(x,y)  of the function f satisfy the equations


    N.B. the of minus; cf. the Cauchy–Riemann equations (

Example.  The function  z1z¯ is antiholomorphic in  {0}.  One has


and thus

Title antiholomorphic
Canonical name Antiholomorphic
Date of creation 2014-11-06 12:07:50
Last modified on 2014-11-06 12:07:50
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Definition
Classification msc 30A99
Synonym antiholomorphic function
Related topic ComplexConjugate