Fourier transform of cosine with square root

Question

In relativistic mechanics, i came across the Fourier transform of the following function : $\cos \left(t \sqrt{x^2+m^2} \right)$ or $e^{it \sqrt{x^2+m^2}}$ ($t$ and $m$ are constants). Is there a way to express the Fourier transform of these functions using special functions ?

Answer 1

This is related. The function $$ f(a,b,x)=\frac{\sin[b\sqrt{a^{2}+x^{2}}]}{\sqrt{a^{2}+x^{2}}} $$ has Fourier transform $$ \hat{f}(a,b,s)= \left\{\begin{array}{cc} \frac{\pi}{2}J_{0}[a\sqrt{b^{2}-s^{2}}], & 0 < s < b \\ 0 & b < s < \infty \end{array}\right. $$ The derivative of $f$ with respect to $b$ is $\cos[b\sqrt{a^{2}+x^{2}}]$, and that should be related to the derivative of $\hat{f}$ with respect to $b$.

Reference: Harry Batemen, Table of Integral Transforms, Vol I, pg. 26 (section 1.7)