In mathematics, pseudoanalytic functions are functions introduced by 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.
Similarities to analytic functions If is not the constant , then the zeroes of are all isolated. Therefore, any analytic continuation of is unique.
Examples Complex constants are pseudoanalytic. Any linear combination with real coefficients of pseudoanalytic functions is pseudoanalytic.
See also
Quasiconformal mapping Elliptic partial differential equations Cauchy-Riemann equations
References
Further reading