Pseudoanalytic function

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:


 * $$u_x=\sigma(x,y)v_y, \quad u_y=-\sigma(x,y)v_x$$

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.