# calculating a complex integral with essential singularity

I want to calculate the integral of the complex function $\dfrac 1 {\sin\frac{1}{z}}$ along the curve $|z|=\frac{1}{5}$. I wanted to use cauchy's residues theorem but the function has an essential singularity at $z=0$ which lies inside the curve so I don't know what to do.

• What is the function $sen$? – Amitai Yuval Jun 25 '17 at 3:53
• $sen$ is $\sin$ in $\texttt{spanish}$. – Felix Marin Jun 25 '17 at 4:03

There are three poles inside the contour $\ds{\braces{z\ \mid\ \verts{z} = 5}}$: single ones at $\ds{\pm\pi}$ and a third order one at $\ds{z = 0}$.
• @AmitaiYuval $\color{#f00}{\mathrm{N0}}$. $z \mapsto 1/z$ makes the integral a 'clockwise' one. So, we have to multiply by $-1$ to rewrite the integral as a 'counterclockwise' one which is required by Cauchy$\ldots$ – Felix Marin Jun 25 '17 at 4:48