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.

Let and let be a real-valued function defined in a bounded domain . If and and are Hölder continuous, then is admissible in . Further, given a Riemann surface , if is admissible for some neighborhood at each point of , is admissible on .

The complex-valued function is pseudoanalytic with respect to an admissible at the point if all partial derivatives of and exist and satisfy the following conditions:

If is pseudoanalytic at every point in some domain, then it is pseudoanalytic in that domain.^[1]

• If is not the constant , then the zeroes of are all isolated.
• Therefore, any analytic continuation of is unique.^[2]

• Complex constants are pseudoanalytic.
• Any linear 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