Pseudoanalytic function

In mathematics, pseudoanalytic functions are functions introduced by Lipman Bers (1950, 1951, 1953, 1956) that generalize analytic functions and satisfy a weakened form of the Cauchy–Riemann equations.

σ ( x , y ) = σ ( z )

be a real-valued function defined in a bounded domain

σ > 0

are Hölder continuous, then

Further, given a Riemann surface

is admissible for some neighborhood at each point of

The complex-valued function

is pseudoanalytic with respect to an admissible

if all partial derivatives of

exist and satisfy the following conditions: If

is pseudoanalytic at every point in some domain, then it is pseudoanalytic in that domain.