3D-Fourier Transform of a delta-function?

Mr.Brown

1. The problem statement, all variables and given/known data
hi
im trying to the integral int(delta(r-b)*exp(ikr)d^3r). but im not really getting anywhere.
I´m trying to integrate over all space in spherical coordinates.
The r part is easy i just do:

delta(r-b)*exp(ikr)r^2*sin(a)*b*dr*da*db -> b^2*exp(ikb*cos(someangle??)*sin(a)*da*db

(sorry that i´m not familiar with tex :( )

I kinda need some help how to do the angular part.
My idea was that the solution should not invole the angles in any sense that´s sure and i think that it´s some kind of trigonometric function but i got no clue how to get somewhere i have some feeling that it´s somethink link sin(bk) * normalization factor but how to get there ?

Any help would be appreciated
thanks :)

Dick

The whole purpose in life of a delta function is to satisfy the condition int(delta(x-a)*f(x))=f(a). So your integral had better come out to be exp(ikb).

Mr.Brown

but im integrating in polar coordinates not in one dimension?
my intetgral (leaving out angular parts!) is int(delta(x-b) * exp(ikx))*r^2 dr isn`t it ?

Dick

If it's a real 3d delta function the coordinates can't matter, can they?

Mr.Brown

hmm yes but it`s a one dimensional delta function in front of a 3d object so you gotta take that into account and take dV=r^2*sin(a)*da*db dont you ?

Dick

Ok, so it's a delta over the coordinate r, not the vector r integrated d^3r. Then what's exp(ikr)? Is r the coordinate r or is it a dot product? If the latter then you have more than one meaning for r in your expression.

Mr.Brown

oh yea sorry for that it think i wrote that somehere in parentesis :)
But youre right i got 2 meanings for one variable sorry :(
you got any idea how to solve that anyways can´t find any solution anywhere even though it seems to be a pretty common problem in stat. mechanics :(

Mr.Brown

:D can be deleted i solved it was really easy just hat to set my coordinate system so that k hat only a z-component :)