Cauchy's Integral

Question

Can we use Cauchy's integral formula to evaluate $\oint_{|z|=1}\frac{zdz}{\sin z}$? If so, how?

Answer 1

As @saulspatz noted, $z=0$ isn't a true singularity of the integrand, nor is any with $|z|<\pi$. The integrand has a Maclaurin series $1+\tfrac16z^2+O(z^4)$; in particular, there's no $\tfrac1z$ term, so the integral is $0$.