- #1
ognik
- 643
- 2
Show that the 3-D FT of a radially symmetric function may be rewritten as a Fourier sin transform
i.e. $ \frac{1}{({2\pi})^{{3}_{2}}} \int_{-\infty}^{\infty}f(r)e^{ik \cdot r} \,d^3x = \frac{1}{k} \sqrt{\frac{2}{\pi}} \int_{-\infty}^{\infty} \left[ rf(r) \right] sin(kr) \,dr $
The example then changes $d^3x$ to $d^3r$ and 'write the integral of this exercise in spherical polar coordinates, make the change of variable $Cos \theta = t$
They then write : $ \frac{1}{({2\pi})^{{3}_{2}}} \int_{0}^{\infty} r^2 f(r) dr \int_{0}^{2\pi} \,d\phi \int_{-1}^{1} e^{ikrt} \,dt $
I don't know this coordinate transform, it doesn't look like the $x = r sin \theta cos \phi$ etc. that I am familiar with, could someone tell me what this transform is please? Also how does the 't' get into the exponent?
i.e. $ \frac{1}{({2\pi})^{{3}_{2}}} \int_{-\infty}^{\infty}f(r)e^{ik \cdot r} \,d^3x = \frac{1}{k} \sqrt{\frac{2}{\pi}} \int_{-\infty}^{\infty} \left[ rf(r) \right] sin(kr) \,dr $
The example then changes $d^3x$ to $d^3r$ and 'write the integral of this exercise in spherical polar coordinates, make the change of variable $Cos \theta = t$
They then write : $ \frac{1}{({2\pi})^{{3}_{2}}} \int_{0}^{\infty} r^2 f(r) dr \int_{0}^{2\pi} \,d\phi \int_{-1}^{1} e^{ikrt} \,dt $
I don't know this coordinate transform, it doesn't look like the $x = r sin \theta cos \phi$ etc. that I am familiar with, could someone tell me what this transform is please? Also how does the 't' get into the exponent?