Landau in the first equation of Über die Gitterpunkte in einem Kreise uses the following formula for the Bessel function of the first kind: $$\frac{1}{2\pi i } \int_{1-\infty}^{1+i\infty} \frac{\mathrm e^{As-B/s}}{s^4} \, \mathrm d s= (A/B)^{3/2} J_3(2 \sqrt{AB})) $$ valid for all $A,B>0$.
Does anyone know of a reference or a proof of that? Since Landau gives no reference or hint I imagine it is either well-known or completely trivial.
First, change the variable in the integral $s=kz$, where $k=\sqrt{B/A}$. You obtain $$\left(A/B\right)^{3/2} \int \frac{\exp{\sqrt{AB}\left(z-1/z\right)}}{z^4}dz.$$ Then deform the contour of integration from the vertical line to $|z|=1$, this is possible since the real part of $(z-1/z)$ is bounded in a simply connected region which contains both contours.
And finally refer to the well known generating function for Bessel functions $$\sum_{-\infty}^\infty J_n(x)z^n=\exp\left(\frac{x}{2}\left(z-1/z\right)\right).$$ Combined with Laurent's theorem this implies that our integral is $J_3(2\sqrt{AB})$.
