# Bateman transform

In the mathematical study of partial differential equations, the **Bateman transform** is a method for solving the Laplace equation in four dimensions and wave equation in three by using a line integral of a holomorphic function in three complex variables. It is named after the English mathematician Harry Bateman, who first published the result in (Bateman 1904).

The formula asserts that if *ƒ* is a holomorphic function of three complex variables, then

is a solution of the Laplace equation, which follows by differentiation under the integral. Furthermore, Bateman asserted that the most general solution of the Laplace equation arises in this way.

## References

- Bateman, Harry (1904), "The solution of partial differential equations by means of definite integrals",
*Proceedings of the London Mathematical Society*,**1**(1): 451–458, doi:10.1112/plms/s2-1.1.451. - Eastwood, Michael (2002),
*Bateman's formula*(PDF), MSRI.

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