Landau in the first equation of [Über die Gitterpunkte in einem Kreise](https://doi.org/10.1007/BF01203524) 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.



infinite integral defining Bessel functions
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 $J\_3(2\sqrt{AB})$.


