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

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

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

If ${\displaystyle f}$ is pseudoanalytic at every point in some domain, then it is pseudoanalytic in that domain.[1]

Similarities to analytic functions

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

Examples

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

See also

• Quasiconformal mapping
• Elliptic partial differential equations
• Cauchy-Riemann equations

References

1. Bers, Lipman (1950), "Partial differential equations and generalized analytic functions" (PDF), Proceedings of the National Academy of Sciences of the United States of America, 36 (2): 130–136, doi:10.1073/pnas.36.2.130, ISSN 0027-8424, JSTOR 88348, MR 0036852, PMC 1063147, PMID 16588958
2. Bers, Lipman (1956), "An outline of the theory of pseudoanalytic functions" (PDF), Bulletin of the American Mathematical Society, 62 (4): 291–331, doi:10.1090/s0002-9904-1956-10037-2, ISSN 0002-9904, MR 0081936

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