- #1
matt_crouch
- 161
- 1
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+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
##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