Evaluating $\int_0^{+\infty}{e^{-2n \sqrt{\pi x}}}cos(2n\sqrt{\pi x})cos(2\pi kx)dx$

Question

Using product to sum formula and substitution $x=t^2$ i got $$\int_0^{\infty}{te^{-at}cos(at+bt^2)}dt=\int_0^{\infty}{te^{-at} \cdot e^{-i(at+bt^2)}}dt=\int_{0}^{\infty}{te^{(-at-iat-ibt^2)}}dt=\int_0^{\infty}{te^{-(mt+nt^2)}}dt$$, assume that m n is greater than 0

Answer 1

Hint

Considering $$I=\int{t\,e^{-(mt+nt^2)}}\,dt$$ perform one integration by parts first. Then, for what is left, complete the square to arrive to a gaussian integral.

If I am not mistaken, you should get something like $$I=-\frac{e^{- (mt+n t^2)}}{2 n}-\frac{\sqrt{\pi }\, m \,e^{\frac{m^2}{4 n}}}{4 n^{3/2}} \text{erf}\left(\frac{m+2 n t}{2 \sqrt{n}}\right)$$