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 + i y {\displaystyle z=x+iy} and let σ ( x , y ) = σ ( z ) {\displaystyle \sigma (x,y)=\sigma (z)} be a real-valued function defined in a bounded domain D {\displaystyle D} . If σ > 0 {\displaystyle \sigma >0} and σ x {\displaystyle \sigma _{x}} and σ y {\displaystyle \sigma _{y}} are Hölder continuous, then σ {\displaystyle \sigma } is admissible in D {\displaystyle D} . Further, given a Riemann surface F {\displaystyle F} , if σ {\displaystyle \sigma } is admissible for some neighborhood at each point of F {\displaystyle F} , σ {\displaystyle \sigma } is admissible on F {\displaystyle F} . The complex-valued function f ( z ) = u ( x , y ) + i v ( x , y ) {\displaystyle f(z)=u(x,y)+iv(x,y)} is pseudoanalytic with respect to an admissible σ {\displaystyle \sigma } at the point z 0 {\displaystyle z_{0}} if all partial derivatives of u {\displaystyle u} and v {\displaystyle v} exist and satisfy the following conditions:
u x = σ ( x , y ) v y , u y = − σ ( x , y ) v x {\displaystyle u_{x}=\sigma (x,y)v_{y},\quad u_{y}=-\sigma (x,y)v_{x}}
If f {\displaystyle f} is pseudoanalytic at every point in some domain, then it is pseudoanalytic in that domain.
Similarities to analytic functions If f ( z ) {\displaystyle f(z)} is not the constant 0 {\displaystyle 0} , then the zeroes of f {\displaystyle f} are all isolated. Therefore, any analytic continuation of f {\displaystyle 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
References
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
