Bateman transform

Summary

The Bateman transform builds solutions of Laplace's equation in four real variables from holomorphic functions of three complex variables. It also gives solutions of the wave equation 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.