Could someone point me to an online resource where I can look up
$$ \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{\cos(\vec k\cdot\vec x)}{\sqrt{|\vec k|^2+m^2}}\,dk_1\,dk_2\,dk_3$$
where $\vec k$ and $\vec x$ are 3-vectors, and $m$ is a constant. Also, if you know how to do it step by step and want to show off your fancy integration, that would also be very helpful. Thanks ;)
On
There is no closed form for the (triple) antiderivative of this function. When taken over all space, this integral does not converge but may be taken as the Fourier transform of $1/\sqrt {k^2+m^2} $. This gives the position space free scalar field propagator, which is given in terms of special functions here. (Under Feynman propagator.)
The integral may be computed by converting it to a four dimensional integral (as the one given on that page is), then applying contour integration and special function techniques. This process is rather involved, however.
Edit: or apply Mark Viola's much more straightforward technique, which I had not seen before :)
The integral diverges, but can be interpreted in terms of tempered distributions. We proceed using formal manipulations.
Rotate the coordinate system of that the polar axis aligns with $\vec x$. Denote $|\vec k|$ by $k$ and $|\vec x|$ by $x$. Then, we can write (in distribution)
$$\begin{align} \int_{-\infty}^\infty \int_{-\infty}^\infty \int_{-\infty}^\infty \frac{\cos(\vec k\cdot\vec x)}{\sqrt{|\vec k|^2+m^2}}\,dk_1\,dk_2\,dk_3&=2\pi \int_0^\infty \int_0^\pi\frac{\cos (kx\cos(\theta))}{\sqrt{k^2+m^2}}\,k^2 \sin(\theta)\,d\theta\,dk\\\\ &=\frac{4\pi}x \int_0^\infty \frac{k\sin(kx)}{\sqrt{k^2+m^2}}\,dk\\\\ &=-\frac{4\pi}x \frac{d}{dx}\int_0^\infty \frac{\cos(kx)}{\sqrt{k^2+m^2}}\,dk\\\\ &=-\frac{4\pi}x \frac{dK_0(|m|x)}{dx}\\\\ &=\frac{4\pi |m| K_1(|m| x)}{x} \end{align}$$