Bateman transform

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 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

ϕ ( w , x , y , z ) = ∮ γ f ( ( w + i x ) + ( i y + z ) ζ , ( i y − z ) + ( w − i x ) ζ , ζ ) d ζ {\displaystyle \phi (w,x,y,z)=\oint _{\gamma }f{\big (}(w+ix)+(iy+z)\zeta ,(iy-z)+(w-ix)\zeta ,\zeta {\big )}\,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, archived from the original on 2013-04-15.
  • Eastwood, Michael (2002), Bateman's formula (PDF), MSRI.


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