Fourier Transform of Window Function

Question

Im trying to fourier transform the following function:

$$W(\textbf{x},R)= \begin{cases} \frac{3}{4\pi R^3},&\text{if r < R}\\ 0,&\text{if r>R} \end{cases}$$ where $r = |x|$.

I've tried to put $x$ in spherical coordinates, so: $$ x = (r\cos(\theta)\sin(\phi),r\sin(\theta)\sin(\phi),r\cos(\theta))$$ and then I have to calculate the following integral: $$\tilde{W}(\textbf{k},R)=\int_{0}^{2\pi} \int^{\pi}_{0} \int_{0}^{R} e^{-i\textbf{k.x}} \ r^2 \sin(\theta) \ dr d\theta \ d\phi $$. which I couldn't do... Is there a simpler way to do that? Or the integral can be solved?

Answer 1

Let $\tilde W(\vec k,R)$ be the function represented by the Fourier integral

$$\tilde W(\vec k,R)=\frac{3}{4\pi R^3}\int_0^{2\pi}\int_0^\pi \int_0^R e^{i\vec k\cdot \vec r}r^2\sin(\theta)\,dr\,d\theta\,d\phi$$

We can rotate our coordinate system so that $\hat z$ aligns with $\vec k$. Then, we can write

$$\begin{align} \tilde W(\vec k,R)&=\frac{3}{4\pi R^3}\int_0^{2\pi}\int_0^\pi \int_0^R e^{i k r\cos(\theta)}r^2\sin(\theta)\,dr\,d\theta\,d\phi\\\\ &=\frac{3}{2 R^3} \int_0^\pi\int_0^Re^{i k r\cos(\theta)}r^2\sin(\theta)\,dr\,d\theta\\\\ & =\frac{3}{2 R^3} \int_0^R r^2\int_0^\pi e^{i k r\cos(\theta)}\sin(\theta)\,d\theta\,dr\\\\ &=\frac{3}{2 R^3} \int_0^R r^2 \left.\left(-\frac{e^{ikr\cos(\theta)}}{ikr}\right)\right|_{0}^{\pi} \,dr\\\\ &=\frac{3}{2 R^3} \int_0^R r\left(\frac{e^{ikr}-e^{-ikr}}{ik}\right)\,dr\\\\ &=\frac{3}{kR^3}\int_0^R r\sin(kr)\,dr\\\\ &=3\left(\frac{\sin(kR)-(kR)\cos(kR)}{(kR)^3}\right) \end{align}$$

Answer 2

The transform of a window function, is the transform of a positive step at beginning and a negative step at the end.
Since you want to calculate an integral of a function depending on the dot product of $\mathbb x$ with another vector$\mathbb v$, over the sphere $|\mathbb x|\leqslant R$, then take the "z" axis in the direction of $\mathbb v$ and the integral will be easily computed.
In the spherical coordinates you gave, note that the $z$ component of $\mathbb x$ should be $r cos\phi$.