Midway through solving a question, I have an intermediate integral:
$$ G(\vec x,t)=\frac{c}{16 \pi^3} \iiint _{\mathbb{R}^3} \frac{\sin (tck)}{k} e^{i \vec{x} . \vec{k}} d^3 \vec k $$
Now what I have to do is:
Integrate of angular variables to bring the expression for the Green's function into the form
$$ G(\vec x,t)=G(x,t)=\frac{c}{8\pi^2} \int _{-\infty}^\infty dk \Big(e^{ik(x-ct)} – e^{ik(x+ct)} \Big) $$
Now converting $\vec k$ to spherical polars, I get:
$$ G(\vec x,t)=\frac{c}{16\pi^3}\iiint_{\mathbb{R^3}}\frac{\sin(tck)}{k} e^{i \vec{x} . \vec{k}} . k^2 \sin^2(\phi) {dk} {d\phi} {d\theta}$$
where $k=|\vec k|$, and $x=|\vec x|$
I'm not sure how to deal with the dot product in the exponent from here.
Best Answer
There is a useful trick here. You could set $$\vec x = (x_1, x_2, x_3)$$ and $$\vec k = (k\cos \phi\cos \theta, k\cos\phi\sin\theta, k\sin\phi),$$ but then you end up with some nasty integrals on $\theta$ and $\phi$ to work out. But since $\vec x$ is constant for the purposes of this integral and the integration is over all space, we can choose a coordinate system with $x$ as the vertical axis ($\phi = \pi/2$). Then $x = (0, 0, x)$ and $$\vec x \cdot \vec k = kx\sin\phi$$