Does the analytical form of the following integral exist?

Question

I have an integral $$\int_0^{2\pi}d\theta\cos(2\theta)e^{-a[1-\cos(\theta-\theta_0)]}.$$ Is there any analytical form for the integral above?

Answer 1

Let $f(a)$ be given by the integral

$$\begin{align} f(a)=\int_0^{2\pi}\cos(2\theta)e^{-a(1-\cos(\theta-\theta_0))}\,d\theta \end{align}$$

Enforcing the substitution $\theta \to \theta+\theta_0$, exploiting the $2\pi$-periodicity of the integrand, and exploiting the oddness of $\sin(2\theta)e^{a\cos(\theta)}$ and the evenness of $\cos(2\theta)e^{a\cos(\theta)}$ yields

$$\begin{align} f(a)&=e^{-a}\int_0^{2\pi}\cos(2(\theta+\theta_0))e^{a\cos(\theta)}\,d\theta\\\\ &=e^{-a}\cos(2\theta_0)\int_0^{2\pi}\cos(2\theta)e^{a\cos(\theta)}\,d\theta-e^{-a}\sin(2\theta_0)\int_0^{2\pi}\sin(2\theta)e^{a\cos(\theta)}\,d\theta\\\\ &=e^{-a}\cos(2\theta_0)\int_0^{2\pi}\cos(2\theta)e^{a\cos(\theta)}\,d\theta\\\\ &=2\pi e^{-a}\cos(2\theta_0) \frac{1}{\pi}\int_0^{\pi}\cos(2\theta)e^{a\cos(\theta)}\,d\theta\\\\ &=2\pi e^{-a}\cos(2\theta_0)I_2(a) \end{align}$$

where $I_2(a)$ is the second order Modified Bessel Function of the first kind.