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.