Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2004 / Analysis / 5

A removable singularity at $a$ means that there is a holomorphic function on the whole disc which agrees with $f$ away from $a$. Equivalently, the Laurent series has no negative powers. The value at $a$ is filled in by its finite limit.

A pole of order $m\ge 1$ means that $(z-a{)}^{m}f(z)$ extends holomorphically to $a$ and has a nonzero value there. Equivalently, the lowest nonzero power in the Laurent expansion is $(z-a{)}^{-m}$.

An essential singularity is an isolated singularity which is neither removable nor a pole; equivalently its Laurent expansion has infinitely many nonzero negative-power terms.

For the final unnumbered classification, the only possible finite singularities are the zeros of the denominator, namely $z=\pm i$ and $z=k\pi$ for integers $k$. At zero,so zero is removable, with extension value one. At each nonzero $k\pi$, $\sin z$ has a simple zero, the other denominator factor is nonzero and the numerator is nonzero. These points are poles of order two, with leading coefficient $(k\pi {)}^2/((k\pi {)}^2+1)$.

At $z=\pm i$, the polynomial denominator has a simple zero. Since $\sin (\pm i)=\pm i\sinh 1\ne 0$ and the numerator is $-1$, these are simple poles. There are no essential singularities on $\mathbb{C}$ and no other finite singularities.