Integrating an exponential function on the hemisphere

Question

I want to calculate the first moment of an exponential function on a hemisphere. The function itself is of the very simple form $$ f(\hat{\omega}) = \exp(\vec{v}\cdot\hat{\omega}), $$ where $\hat{\omega}$ is a unit vector on the (hemi)sphere and $\vec{v}$ is some arbitrary vector. The integral I want to calculate is $$ \int_{\Omega^+} (\hat{z}\cdot\hat{\omega}) \ f(\hat{\omega}) \,\mathrm{d}^2\hat{\omega} = \int_{\Omega^+} (\hat{z}\cdot\hat{\omega}) \ \exp(\vec{v}\cdot\hat{\omega}) \,\mathrm{d}^2\hat{\omega}, $$ where $\Omega^+$ is the upper hemisphere (i.e. $\hat{z}\cdot\hat{\omega}\ge0$). Writing this in standard spherical coordinates gives $$ \int_0^{\pi/2}\mathrm{d}\theta \int_0^{2\pi}\mathrm{d}\phi\ \sin\theta\ \cos\theta\ \exp{(a\sin\theta \cos\phi + b\cos\theta)}, $$ where I chose the frame such that $\vec{v}=(a,0,b)$. This smells like some kind of Bessel contraption, but I can't quite seem to solve it. (Maybe with some funky contour integral magic?)