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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]

Similarities to analytic functions

Examples

  • Complex constants are pseudoanalytic.
  • Any linear combination with real coefficients of pseudoanalytic functions is pseudoanalytic.[1]

See also

References

Further reading

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