Pseudoanalytic function explained

In mathematics, pseudoanalytic functions are functions introduced by that generalize analytic functions and satisfy a weakened form of the Cauchy–Riemann equations.

Definitions

Let

z = x + iy and let \sigma(x,y) = \sigma(z) be a real-valued function defined in a bounded domain D . If \sigma> 0 and \sigma _{x} and \sigma _{y} are Hölder continuous, then \sigma is admissible in D . Further, given a Riemann surface F , if \sigma is admissible for some neighborhood at each point of F , \sigma is admissible on F .

The complex-valued function

f(z) = u(x,y) + iv(x,y) is pseudoanalytic with respect to an admissible \sigma at the point z _{0} if all partial derivatives of u and v exist and satisfy the following conditions:u _{x=\sigma(x,y)v} _{y,} u _{y=-\sigma(x,y)v} _{x}

If

f is pseudoanalytic at every point in some domain, then it is pseudoanalytic in that domain.

Similarities to analytic functions

• If

f(z) is not the constant 0 , then the zeroes of f are all isolated.

• Therefore, any analytic continuation of

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

Further reading

• Book: Kravchenko . Vladislav V. . Applied pseudoanalytic function theory . 2009 . Birkhauser . 978-3-0346-0004-0.