Hello everybody I am trying to solve this integral. I show you how far I 've gone. $\displaystyle\int^{\infty}_{-\infty} \frac {e^{-i\vec{k}.\vec{x}}e^{-ik^{\circ}x_{\circ}}\delta(k^{\circ})}{k^2-\mu^2}d^2kdk^{\circ}$. Taking to plane polar coordinates
$=\displaystyle\int^{\infty}_{0}\displaystyle\int^{2\pi}_{0}\displaystyle\int^{\infty}_{-\infty}\frac {e^{-i|\vec{k}| |\vec{x}|\cos\theta}e^{-ik^{\circ}x_{\circ}}\delta(k^{\circ})}{k^{\circ 2} -|\vec{k}|^2-\mu^2}|\vec{k}|dkd\theta dk^{\circ}$.
Where $k^2=k^{\circ 2} -|\vec{k}|^2 $.
Now solving delta function we get.
$=\displaystyle\int^{\infty}_{0}\displaystyle\int^{2\pi}_{0}\frac {e^{-i|\vec{k}| |\vec{x}|\cos\theta}}{-|\vec{k}|^2-\mu^2}|\vec{k}|dkd\theta$.
Now bring in Bessel function
$=-\sum_{n=0}^\infty (-1)^n \frac{2\pi}{n!n!}\displaystyle\int^{\infty}_{0}\left(\frac{|\vec{k}||\vec{x}|}{2}\right)^{2n}\frac {1}{|\vec{k}|^2+\mu^2}|\vec{k}|dk\\$.
Nowwwww I don't know what to do.........