For the function $exp(\frac{1}{sinz})^2)$ i have to find if it has an essential, removable or a Pole singularity.
My Idea was to do the Laurent series of it and I got $\sum_{n=0}^{\infty}\frac{1}{(sinz)^{2n}*n!}$ and I thought that since here we have infinitely many negative powers than it has to be an essential singularity.
Would this be correct?
Thanks
The only singular points are $k\pi.$ It suffices to study the point $z=0,$ as the function has period $\pi.$ Assume $|z|<\pi.$ Then $$g(z):={1\over z^2}-{1\over \sin ^2 z} ={\sin z-z\over z\sin^2 z}\,{\sin z+z \over z}={\sin z-z\over z^3}\,{z^2\over \sin^2z}\,\left ({\sin z \over z}+1\right )$$The function $g(z)$ is holomorphic in $|z|<\pi,$ as $\displaystyle \lim_{z\to 0}g(z)=-{1\over 3}.$ We have $$e^{g(z)}e^{1/\sin^2z}= e^{1/z^2}$$ The function $e^{g(z)}$ is holomorphic in $|z|<\pi.$ As the point $0$ is an essential singularity of $e^{1/z^2}$ it must be an essential singularity of $e^{1/\sin^2z}.$