Cauchy's Integral

Question

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

The singularity at $z=0$ is removable. – saulspatz Jan 22 '21 at 22:09 — saulspatz, Jan 22 '21 at 22:09
@KentaS Not really; $\pi\mapsto?$ – José Carlos Santos Jan 22 '21 at 22:25 — José Carlos Santos, Jan 22 '21 at 22:25

score 1 · Answer 1 · answered Jan 22 '21 at 22:14

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 $0$.

answered Jan 22 '21 at 22:14

J.G.

115,835

Cauchy's Integral

1 Answers1