# Bateman transform

From HandWiki

__: Method for solving the Laplace equation in four dimensions__

**Short description**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

- [math]\displaystyle{ \phi(w,x,y,z) = \oint_\gamma f\left((w+ix)+(iy+z)\zeta,(iy-z)+(w-ix)\zeta,\zeta\right)\,d\zeta }[/math]

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, http://plms.oxfordjournals.org/cgi/reprint/s2-1/1/451. - Eastwood, Michael (2002),
*Bateman's formula*, MSRI, http://www.msri.org/ext/concepts/eastwood3.pdf.

Original source: https://en.wikipedia.org/wiki/Bateman transform.
Read more |