Interesting definite integral involving exp and trig I'm trying to evaluate the following integrals:
$$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \cos(\phi) d\phi$$
$$\int_0^{2\pi} e^{\kappa \cos(\phi - \mu)} \sin(\phi) d\phi$$
for which I want to find an easily computable function. This may be either a closed form, or something in terms of special functions that are available in most scientific computing libraries (e.g. scipy.special).
I found the following result on wikipedia, which may be useful. 
$$\int_0^{2\pi} e^{x \cos(\theta)} d\theta = 2\pi I_0(x),$$
where $I_0(x)$ is the modified Bessel function of the first kind, of order 0.
I want to apologize in advance if this is a basic question; I have not had formal training in advanced calculus (beyond highschool), except for a few online lectures and wikipedia reading.
Any help is greatly appreciated.
 A: Consider $$f(x):=\int_0^{2\pi} e^{x \cos(\phi)}\;d\phi=2\pi I_0(x)$$
since $I'_0(x)=I_1(x)$ and using derivation under the integral sign this becomes :
$$f'(x)=\int_0^{2\pi} e^{x \cos(\phi)}\;\cos(\phi) \;d\phi=2\pi I'_0(x)=2\pi I_1(x)$$
Let's rewrite your first integral (using the substitution $\theta:=\phi - \mu$) :
\begin{align}
I_c:&=\int_{-\mu}^{2\pi-\mu} e^{\kappa \cos(\theta)} \cos(\theta+\mu)\;d\theta\\
&=\int_0^{2\pi} e^{\kappa \cos(\theta)} \cos(\theta+\mu)\;d\theta\\
&=\cos(\mu)\int_0^{2\pi} e^{\kappa \cos(\theta)} \cos(\theta)\;d\theta-\sin(\mu)\int_0^{2\pi} e^{\kappa \cos(\theta)} \sin(\theta)\;d\theta\\
&=2\pi\cos(\mu)I_1(\kappa)+\sin(\mu)\int_0^{2\pi} e^{\kappa \cos(\theta)} \;d\cos(\theta)\\
&=2\pi\cos(\mu)I_1(\kappa)+\sin(\mu)\frac{e^{\kappa\cos(2\pi)}-e^{\kappa\cos(0)}}{\kappa}\\
I_c&=2\pi\cos(\mu)\;I_1(\kappa)
\end{align}
The same way I got the second integral as :  
$\qquad\qquad\qquad I_s=2\pi\sin(\mu)\;I_1(\kappa)$
For interesting properties and rewriting of the modified Bessel function $I_1$ see the excellent references available : 


*

*Abramowitz & Stegun (with a picture of $I_1$ and numerical approximations page $378$)

*DLMF (kind of online successor of A & S)

*Wolfram functions (very large database of information).

