I have a parameter $\beta\geq 0$ and the following integral \begin{equation} I(\beta)=\int_{-\pi}^{\pi}\frac{\mathrm{d}k}{2\pi}\, \ln(1+e^{-\beta \cos k}) \end{equation} And I'm trying to figure out the asymptotic for $\beta \rightarrow 0$ and $\beta$ very large.
The first seems easy using a Taylor expansion, yet the second one is a bit more complicated because of the sign of $\cos(k)$ changing within the interval of integration.
