$e^{-c\sin \theta}$ integral evaluation

Question

I want to compute the following integral: $$ I=\int_{\pi/3}^{2\pi/3}d\theta \, e^{-c \sin \theta}, $$ for some generic constant $c$ — any ideas?

Answer 1

Hint:

Using the Taylor development of the exponential,

$$I(c)=2\int_0^{\pi/6}e^{-c\cos\theta}d\theta=2\sum_{k=0}^\infty(-1)^k \left[\int_0^{\pi/6}\cos^{k}\theta\,d\theta\right]\frac{c^{k}}{k!}.$$

The integrals can be computed using the binomial development of $(e^{i\theta}+e^{-i\theta})^{k}$ and will result in a linear combination of binomial numbers. (WA gives an expression in terms of semi-factorials and an hypergeometric function.)

It is also possible to develop directly in terms of $\theta$ rather than $\cos\theta$. Probably equally difficult.