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.

See also

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

References

Further reading

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* {{Citation, last1=Bers, first1=Lipman, title=Theory of pseudo-analytic functions, url=https://books.google.com/books?id=79dWAAAAMAAJ, year=1953, publisher=Institute for Mathematics and Mechanics, New York University, New York, mr=0057347

Complex analysis
Partial differential equations
Types of functions