Ampère's force law: Difference between revisions

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In the [[mathematics|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 number|complex variables]]. It is named after the English mathematician [[Harry Bateman]], who first published the result in {{harv|Bateman|1904}}.


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The formula asserts that if ''ƒ'' is a holomorphic function of three complex variables, then
 
:<math>\phi(w,x,y,z) = \oint_\gamma f\left((w+ix)+(iy+z)\zeta,(iy-z)+(w-ix)\zeta,\zeta\right)\,d\zeta</math>
 
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==
*{{citation|doi=10.1112/plms/s2-1.1.451|last=Bateman|first=Harry|authorlink=Harry Bateman|title=The solution of partial differential equations by means of definite integrals|journal=[[Proceedings of the London Mathematical Society]]|year=1904|volume=1|issue=1|pages=451–458|url=http://plms.oxfordjournals.org/cgi/reprint/s2-1/1/451}}.
*{{citation|first=Michael|last=Eastwood|authorlink=Michael Eastwood|title=Bateman's formula|url=http://www.msri.org/ext/concepts/eastwood3.pdf|year=2002|publisher=[[MSRI]]}}.
 
[[Category:Harmonic analysis]]
[[Category:Integral geometry]]
[[Category:Partial differential equations]]
[[Category:Several complex variables]]
 
 
{{mathanalysis-stub}}

Revision as of 18:38, 2 January 2014

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

ϕ(w,x,y,z)=γf((w+ix)+(iy+z)ζ,(iyz)+(wix)ζ,ζ)dζ

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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