Generalization of analytic functions
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.
Definitions
Let and let be a real-valued function defined in a bounded domain . If and and are Hölder continuous, then is admissible in . Further, given a Riemann surface , if is admissible for some neighborhood at each point of , is admissible on .
The complex-valued function is pseudoanalytic with respect to an admissible at the point if all partial derivatives of and exist and satisfy the following conditions:
If is pseudoanalytic at every point in some domain, then it is pseudoanalytic in that domain.[1]