Evaluate the integrals $\int_\gamma z^ndz$ for all integers n. Here $\gamma$ is any circle not containing the origin.
The answer to this problem is extremely difficult.
$$ \int \limits_{\gamma }\frac{dz}{z} \text{=}\int \limits^{\pi }_{\theta =-\pi }\frac{ire^{i\theta }}{a+re^{i\theta }} d\theta =ir\int \limits^{2\pi }_{\theta =0}\frac{a\cos \left( \theta \right) +r+ia\sin \left( \theta \right) }{a^{2}+2ar\cos \left( \theta \right) +r^{2}} d\theta =ir\int \limits^{\pi }_{\theta =-\pi }\frac{a\cos \left( \theta \right) +r}{a^{2}+2ar\cos \left( \theta \right) +r^{2}} d\theta $$ . Because $ \sin (-\theta) =-\sin (\theta) $. Looks like this. I am just wondering if there is any simpler way to solve it. thanks.