The Anger-Weber function is defined by $$ A_{\nu}(z) = \int_0^\infty \exp\bigl(-\nu t - z \sinh(t)\bigr) \mathrm{d}t $$ where $\nu, z \in \mathbb{C}$ with $\Re(z) > 0$.
I am not able to numerically evaluate this integral.
But there is an elementary relation involving this function, the Bessel function of the first kind $J$, and the Anger function $\mathbf{J}$. I'm able to numerically evaluate $J$ and $\mathbf{J}$, and thanks to this elementary relation, I can get $A_\nu(z)$ except when $\nu$ is an integer (because there's a division by $\sin(\nu\pi)$).
Therefore I'm looking for a way to evaluate $A_\nu(z)$ for the special case when $\nu$ is an integer.