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 $z=x+iy$ 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