pole and essential singularity in the same point

Question

In this case

in $0%$ i have an essential singularity by $\sin(\frac{1}{s})$ but i have a ""pole"" too ( the denominator of the fractions $\frac{(...)\sin(...)}{**S**}$) , so in this case is this a pole or is this an essential singualrity and why ?

Answer 1

This is an essential singularity.

An essential singularity is characterized by an infinite number of negative degree terms in its Laurent series expansion. So the $\frac{1}{s}$ contribution, which has only one such term (namely $\frac{1}{s}$ itself), plays no role.

Answer 2

It's an essential singularity. That's because the Laurent expansion at $0$ has infinitely many nonzero terms in the singular part (the terms with negative powers of $z$).