Pseudoanalytic function

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 $\sigma (x,y)=\sigma (z)$ be a real-valued function defined in a bounded domain $D$. If $\sigma >0$ and $\sigma _{x}$ and $\sigma _{y}$ are Hölder continuous, then $\sigma$ is admissible in $D$. Further, given a Riemann Surface $F$, if $\sigma$ is admissible for some neighborhood at each point of $F$, $\sigma$ is admissible on $F$.

The complex-valued function $f(z)=u(x,y)+iv(x,y)$ is pseudoanalytic with respect to an admissible $\sigma$ at the point $z_{0}$ if all partial derivatives of $u$ and $v$ exist and satisfy the following conditions:

$u_{x}=\sigma (x,y)v_{y},\quad u_{y}=-\sigma (x,y)v_{x}$

If $f$ is pseudoanalytic at every point in some domain, then it is pseudoanalytic in that domain.

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.

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" (PDF), 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, JSTOR 88213, MR 0044006, PMC 1063297, PMID 16588987
• Bers, Lipman (1953), Theory of pseudo-analytic functions, Institute for Mathematics and Mechanics, New York University, New York, MR 0057347