I'm studying from notes taken in class, the context of which is not relevant to the question. The problem is simply as follows. I have this expression:
$$f(y)=\frac{1}{2\pi}\int_{\phi=0}^{2\pi}e^{y\cdot\cos(\phi-\phi_0)}\,\mathrm{d}\phi$$
and in the notes I read that immediately:
$$f(y)=I_0(y)$$
This is not so clear to me, because I know that the definition of the $0$-th order modified Bessel function is:
$$I_0(y)=\frac{1}{\pi}\int_{t=0}^{\pi}e^{y\cdot\cos(t)}\,\mathrm{d}t$$
and I can't find a simple change of variable that transforms $f(y)$ into $I_0(y)$.