How to calculate this complex integral by Cauchy integral formula?

Question

I met an integral like this: $\oint_{|z|=1}\frac{dz}{z^2 sin(z)}$. I know that it can be worked out by using Residue theorem. How can we calculate it using other methods, such as Cauchy integral formula? (In fact, this integral is an exercise in a chapter before the Residue theorem in a book on complex analysis. )

Answer 1

HINT:

Write $\frac{1}{z^2\sin(z)}=\frac{z\csc(z)}{z^3}$ and note that $f(z)=z\csc(z)=\frac{z}{\sin(z)}$ is holomorphic on $|z|\le 1$ (with a removeable singularity at $z=0$).

Then, apply Cauchy's Integral Formula

$$f''(0)=\frac{2!}{2\pi i}\oint_{|z|=1}\frac{f(z)}{z^3}\,dz$$

with $f(z)=\frac{z}{\sin(z)}$.

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NOTE: Calculating $f''(0)$ will take a bit of work but is facilitated by using series expansion. The result is $f''(0)=\frac13$.