I recently have a problem as follows
$\int_0^\infty \cos \bigl[ { k \cdot t \over {\sqrt{1 + k^2}} } \bigr] \cdot \cos \bigl[ k \cdot x \bigr] \, dk$
Here, x and t are spatial and temporal constant. I’m trying to find the analytic form for the root. I’ve tried to (and considered) find it via complex analysis(Residue theorem), Riemann–Lebesgue lemma, integration table, quadratures, etc…, however, it failed.
Thank you for your advice for comments for that and the answers in advance. Many thanks indeed.