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, Bibcode:1950PNAS...36..130B, 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, Bibcode:1951PNAS...37...42B, 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