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
$w=i\sqrt{2}z+x\cos\vartheta +iy\sin(\vartheta)$
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 tried transforming to the new coordinates through
$$x=\hat{d}(w+\tilde{w}),$$ $$y=i\hat{e}(w+\tilde{w}),$$ $$z=i\sqrt{2}(w-\tilde{w})$$
but couldnt really get anywhere from here... 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 =]