---
tag: KR5Q7WMAVC
kind: concept
name: Bateman transform
msc: [35C15, 31B05, 35L05]
specializes: [YCK709QB06]
sources:
  - url: https://en.wikipedia.org/wiki/Bateman_transform
same_as:
  wikidata: Q4868770
  wikipedia: Bateman transform
---

## Summary

The Bateman transform builds solutions of [[Laplace's equation|S8RG6N1PRT]] in four real variables from holomorphic functions of three complex variables.
It also gives solutions of the [[wave equation|5EHD2XBQDG]] in three space dimensions.
The solution is a contour integral in one complex variable $\zeta$ of a holomorphic function evaluated at two linear expressions in the coordinates.
It is named after Harry Bateman and extends Whittaker's integral formula for harmonic functions in three variables.
It is a forerunner of the Penrose transform of twistor theory.

## Explanations

### L1 human

Holomorphic functions of one variable give harmonic functions of two variables for free: their real parts.
The Bateman transform does something similar one level up.
It plugs carefully chosen combinations of four real coordinates into a holomorphic function and integrates along a loop.
The combinations are chosen so that the Laplacian of the result vanishes automatically, whatever the holomorphic function is.