In the
mathematical study of
partial differential equations, the Bateman transform is a method for solving the
Laplace equation
In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as
\nabla^2\! f = 0 or \Delta f = 0,
where \Delta = \n ...
in four dimensions and
wave equation in three by using a
line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; ''contour integral'' is used as well, a ...
of a
holomorphic function in three
complex variables. It is named after the English mathematician
Harry Bateman
Harry Bateman FRS (29 May 1882 – 21 January 1946) was an English mathematician with a specialty in differential equations of mathematical physics. With Ebenezer Cunningham, he expanded the views of spacetime symmetry of Lorentz and Poinca ...
, who first published the result in .
The formula asserts that if ''ƒ'' is a holomorphic function of three complex variables, then
:
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
*.
*.
Harmonic analysis
Integral geometry
Partial differential equations
Several complex variables
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