Pseudoanalytic function

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