Bateman transform

Short description: Method for solving the Laplace equation in four dimensions

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\big((w + ix) + (iy + z)\zeta, (iy - z) + (w - ix)\zeta, \zeta\big) \,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 .