---
tag: Y1MH7WPRDZ
kind: concept
name: Pseudoanalytic function
msc: [30G20]
sources:
  - url: https://en.wikipedia.org/wiki/Pseudoanalytic_function
same_as:
  wikidata: Q7254510
  wikipedia: Pseudoanalytic function
---

## Summary

A pseudoanalytic function is a complex function that is "holomorphic relative to a pair of basis functions" $(F, G)$ instead of relative to the pair $(1, i)$.
The notion was introduced by Lipman Bers to extend the theory of [[holomorphic functions|0NAHXR1FRH]] to solutions of first-order elliptic systems in the plane.
For smooth generating pairs, pseudoanalytic functions satisfy equations of the form $w_{\bar z} = a w + b \bar w$, and many theorems of complex analysis carry over to them.

## Explanations

### L1 human

Every complex number $w$ can be written as $\varphi \cdot 1 + \psi \cdot i$ with $\varphi, \psi$ real, and $w$ is holomorphic when these real coefficients vary in a special way.
A pseudoanalytic function replaces the fixed pair $(1, i)$ by a pair of functions $(F, G)$ that are independent over the reals at each point.
One writes $w = \varphi F + \psi G$ and asks the real coefficients to satisfy the analogue of the Cauchy–Riemann condition.
With $F = 1$ and $G = i$ this is exactly ordinary holomorphy.