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.
[Math] calculating a complex integral with essential singularity
complex-analysis
Best Answer
$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,} \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace} \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack} \newcommand{\dd}{\mathrm{d}} \newcommand{\ds}[1]{\displaystyle{#1}} \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,} \newcommand{\ic}{\mathrm{i}} \newcommand{\mc}[1]{\mathcal{#1}} \newcommand{\mrm}[1]{\mathrm{#1}} \newcommand{\pars}[1]{\left(\,{#1}\,\right)} \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}} \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,} \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}} \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$ \begin{align} &\oint_{\verts{z} = 1/5}{\dd z \over \sin\pars{1/z}} \,\,\,\stackrel{z\ \mapsto\ 1/z}{=}\,\,\, \oint_{\verts{z} = 5}{\dd z \over z^{2}\sin\pars{z}} \\[5mm] & = 2\pi\ic\braces{% \lim_{z \to 0}{1 \over 2!}\totald[2]{}{z}\bracks{z \over \sin\pars{z}} + \lim_{z \to -\pi} \bracks{z + \pi \over z^{2}\sin\pars{z}} + \lim_{z \to \pi}\bracks{z - \pi \over z^{2}\sin\pars{z}}} \\[5mm] & = 2\pi\ic\pars{{1 \over 6} + {-1\phantom{^{2}} \over \pi^{2}} + {-1\phantom{^{2}} \over \pi^{2}}} = \bbx{\pars{{1 \over 3}\,\pi - {4 \over \pi}}\ic} \end{align}