In mathematics,pseudoanalytic functions are functions introduced byLipman Bers (1950,1951,1953,1956) that generalizeanalytic functions and satisfy a weakened form of theCauchy–Riemann equations.

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${\displaystyle {\sigma }_y}$ areHölder continuous, then${\displaystyle \sigma }$ is admissible in${\displaystyle D}$. Further, given aRiemann 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:

${\displaystyle u_x=\sigma (x,y)v_y,u_y=-\sigma (x,y)v_x}$

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

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

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

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

• 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

• Complex analysis
• Partial differential equations
• Types of functions

• Articles with short description
• Short description matches Wikidata