antiholomorphic

Antiholomorphic (or anti-holomorphic) describes a class of functions or maps that are holomorphic with respect to the conjugate complex structure. In one complex variable, a function f defined on an open set Ω ⊂ C is antiholomorphic if it is differentiable with respect to the conjugate variable z̄, equivalently ∂f/∂z = 0. This means, up to domain considerations, f can be written as f(z) = g(z̄) for some holomorphic function g on the conjugate domain. A canonical example is the complex conjugation map z ↦ z̄. More generally, any function of the form f(z) = overline{g(z)} with g holomorphic (or f(z) = h(z̄) with h holomorphic) is antiholomorphic.

Antiholomorphic functions are orientation-reversing conformal (anti-conformal) on domains in the plane. They invert the roles of z and z̄ in the Cauchy–Riemann equations, which accounts for the reversal of orientation under such maps. Basic algebraic properties include that the composition of two antiholomorphic maps is holomorphic, while the composition of a holomorphic and an antiholomorphic map is antiholomorphic.

In higher dimensions, antiholomorphic maps are defined similarly in the context of several complex variables or complex manifolds. A map f between complex manifolds is antiholomorphic if it is holomorphic with respect to the opposite (conjugate) complex structures; equivalently, its differential is complex-antilinear.

Antiholomorphic concepts appear in complex analysis, differential geometry, and Teichmüller theory, and they provide a natural counterpart to holomorphic maps, with the complex conjugation map serving as the simplest standard example.