In the problem there are two integrals, and one is asked to evaluate them by taking an integral over a unit circle of some chosen function. The integrals are
$$\int_0^{2\pi}e{^{\sin n\theta}}\cos (\theta - \cos n\theta)\ d\theta$$ and $$\int_0^{2\pi}e{^{\sin n\theta}}\sin (\theta - \cos n\theta)\ d\theta$$
Ok, they look quite similar to the Bessel function which is $$J_n(\alpha) = \frac{1}{2\pi}\int_{-\pi}^{\pi}\cos(\alpha \sin\theta - n\theta)\ d\theta$$ which in turn is a factor involved in the Laurent series expansion of an exponential function that is $$\exp\bigg({\frac{\alpha}{2}\big(z-\frac{1}{z}\big)}\bigg)=\sum \limits _{n = -\infty} ^{n = +\infty} J_n(\alpha)z^n$$ We also know that in the Laurent series for $f(z)$ analytic inside some circle around a singularity $$f(z) = \sum \limits _{n = -\infty} ^{n = +\infty} a_n(z-z_0)^n \ , a_n = \frac{1}{2\pi i}\oint_C\frac{f(\zeta)\ d\zeta}{(\zeta-z_0)^{n+1}}$$ So my guess is I am looking for a function that when converted to a function over a unit circle will be of the form $e^{\sin n\theta}$ and that this function will be similar to the exponential mentioned above. But I just can't figure out how to obtain that function or even go from $(\theta - \cos n\theta)$ to $(n\theta - \cos\theta)$ in the integrand (a substitution would alter integration limits).
Any hints?