Pseudoanalytic function

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 [math]\displaystyle{ z=x+iy }[/math] and let [math]\displaystyle{ \sigma(x,y)=\sigma(z) }[/math] be a real-valued function defined in a bounded domain [math]\displaystyle{ D }[/math]. If [math]\displaystyle{ \sigma\gt 0 }[/math] and [math]\displaystyle{ \sigma_x }[/math] and [math]\displaystyle{ \sigma_y }[/math] are Hölder continuous, then [math]\displaystyle{ \sigma }[/math] is admissible in [math]\displaystyle{ D }[/math]. Further, given a Riemann surface [math]\displaystyle{ F }[/math], if [math]\displaystyle{ \sigma }[/math] is admissible for some neighborhood at each point of [math]\displaystyle{ F }[/math], [math]\displaystyle{ \sigma }[/math] is admissible on [math]\displaystyle{ F }[/math].

The complex-valued function [math]\displaystyle{ f(z)=u(x,y)+iv(x,y) }[/math] is pseudoanalytic with respect to an admissible [math]\displaystyle{ \sigma }[/math] at the point [math]\displaystyle{ z_0 }[/math] if all partial derivatives of [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] exist and satisfy the following conditions:

[math]\displaystyle{ u_x=\sigma(x,y)v_y, \quad u_y=-\sigma(x,y)v_x }[/math]

If [math]\displaystyle{ f }[/math] is pseudoanalytic at every point in some domain, then it is pseudoanalytic in that domain.

Similarities to analytic functions

• If [math]\displaystyle{ f(z) }[/math] is not the constant [math]\displaystyle{ 0 }[/math], then the zeroes of [math]\displaystyle{ f }[/math] are all isolated.
• Therefore, any analytic continuation of [math]\displaystyle{ f }[/math] 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

Further reading

• Kravchenko, Vladislav V. (2009). Applied pseudoanalytic function theory. Birkhauser. ISBN 978-3-0346-0004-0. 
• Bers, Lipman (1951), "Partial differential equations and generalized analytic functions. Second Note", Proceedings of the National Academy of Sciences of the United States of America 37 (1): 42–47, doi:10.1073/pnas.37.1.42, ISSN 0027-8424, PMID 16588987, PMC 1063297, Bibcode: 1951PNAS...37...42B, http://www.pnas.org/content/37/1/42.full.pdf 
• Bers, Lipman (1953), Theory of pseudo-analytic functions, Institute for Mathematics and Mechanics, New York University, New York, https://books.google.com/books?id=79dWAAAAMAAJ