# Bateman transform

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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 Template:Harv.

The formula asserts that if ƒ is a holomorphic function of three complex variables, then

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

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

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