Evaluating the integral of $1+z+1/\tan z$ over a circle

Question

I am a beginner and I want to learn how to solve these kind of integrals:

$$\int_{|z|= \pi/4}\left(1+z+\frac{1}{\tan z}\right)\,dz$$

So should I divide it in three integrals, calculate each integral and use the residue theorem?

Answer 1

You can split into two, rather than three. Since $1+z$ is holomorphic everywhere, its integral over a closed contour is zero. This leaves you just with the $1/\tan z $. I recommend writing it as $\frac{\cos z}{\sin z}$ and computing the residue at a pole $a$ as $$ \frac{\cos z}{(\sin z)'}\bigg|_{z=a} $$