So It is well known that the 2D solution to the Laplace equation can be obtained by changing to complex coordinates ##u=x+iy## and ##v=x-iy##. This can be extended to n dimensions as long as the complex coordinates chosen also solve the Laplace equation. For example in 3D(adsbygoogle = window.adsbygoogle || []).push({});

##w=i\sqrt{2}z+xcos\vartheta +iysin(\vartheta)##

along with the complex conjugate of w. This allows the 3D laplace equation to be solved in the same way as the 2D case. I.e the solution of

##\nabla^{2}f=0## can be found through

##f=g(w)+h(\tilde{w})##

A general solution to the Laplace equation is given by

##f=\int f(w,\vartheta)d\vartheta##

this is called Whittaker's solution and from this we can obtain

##f=\frac{2\pi x}{(x^{2}+y^{2}+z^{2})^{3/2}}##

which is a solution to the Laplace equation...

Ok so my question is I am unsure how to show that they are equivalent?

Whittaker's solution must be able to be expressed as

##f=\int f(w,\vartheta)d\vartheta=g(w)+h(\tilde{w})=\frac{2\pi x}{(x^{2}+y^{2}+z^{2})^{3/2}}##

I am sure there is a simple explanation? probably involving a taylor expansion or something? but I am not sure how to get there...I believe these all have some context and relation to twistor theory and the like but before i delve cohmology etc I am hoping for an easier solution =]

Matt

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# Whittaker's solution and separable variables

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