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

\phi (w,x,y,z)=\oint _{\gamma }f\left((w+ix)+(iy+z)\zeta ,(iy-z)+(w-ix)\zeta ,\zeta \right)\,d\zeta

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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