Pseudoanalytic function

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Short description: 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 z=x+iy and let σ(x,y)=σ(z) be a real-valued function defined in a bounded domain D. If σ>0 and σx and σy are Hölder continuous, then σ is admissible in D. Further, given a Riemann surface F, if σ is admissible for some neighborhood at each point of F, σ is admissible on F.

The complex-valued function f(z)=u(x,y)+iv(x,y) is pseudoanalytic with respect to an admissible σ at the point z0 if all partial derivatives of u and v exist and satisfy the following conditions:

ux=σ(x,y)vy,uy=σ(x,y)vx

If f is pseudoanalytic at every point in some domain, then it is pseudoanalytic in that domain.[1]

Similarities to analytic functions

  • If f(z) is not the constant 0, then the zeroes of f are all isolated.
  • Therefore, any analytic continuation of f is unique.[2]

Examples

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

See also

References

Further reading