I am looking for a closed form or special integral representation of:
$ I(a,b, \phi_0) = \int_0^{2\pi} e^{a\phi + b\cos(\phi - \phi_0)} ~d\phi \qquad a,b \in \mathbb{R}, \phi_0 \in [0,2\pi) $
I know, that for $a=0$ one can use the Modified Bessel function of first kind $I(0,b,\phi_0) = 2\pi I_0(b)$.