Pseudoanalytic function

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 ${\displaystyle z=x+iy}$ 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 ${\displaystyle \sigma _{y}}$ are Hölder continuous, then ${\displaystyle \sigma }$ is admissible in ${\displaystyle D}$. Further, given a Riemann 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:

${\displaystyle u_{x}=\sigma (x,y)v_{y},\quad u_{y}=-\sigma (x,y)v_{x}}$

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

Similarities to analytic functions

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

Examples

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

See also

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