Im trying to proof or disprove the following claim:
If $f(z)$ and $g(z)$ are holomorphic in an annulus $0 < |z − z(\beta)| < R$ and $f$ has an essential singularity at $z(\beta)$ and $g$ has a pole of order $n$ at $z(\beta)$, then $f(z) · g(z)$ has an essential singularity at $z(\beta)$.
The only thing I can think of is that if I multiply the power serieses (Laurent serieses) around $z(\beta)$ of both $f$ and $g$ then I would still have an infinite number of cooeficients. Is is sums the whole claim? because it seems to simple, am I missing something?