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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.¹

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.²

Examples

• Complex constants are pseudoanalytic.
• Any linear combination with real coefficients of pseudoanalytic functions is pseudoanalytic.³

See also

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

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

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 http://www.pnas.org/content/36/2/130.full.pdf ↩

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 https://www.ams.org/journals/bull/1956-62-04/S0002-9904-1956-10037-2/S0002-9904-1956-10037-2.pdf ↩

3. 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 http://www.pnas.org/content/36/2/130.full.pdf ↩