MHB [Complex Analysis] Singularity in product of analytic functions

[ludwig1]

Suppose f,g:ℂ→ℂ are analytic with singularities at z=0. I was wondering whether f(z)^2 or f(z)g(z) will have a singularity at z=0? For each, can you give me a proof or a counterexample?

 

[S.G. Janssens]

Suppose f,g:ℂ→ℂ are analytic with singularities at z=0. I was wondering whether f(z)^2 or f(z)g(z) will have a singularity at z=0? For each, can you give me a proof or a counterexample?

I think the most staightforward (but perhaps not the most handy?) approach would be to just consider Laurent series for $f$ and $g$ on the same annulus centered at the origin. What do you know about the coefficients in the product of two such series?

Assuming that "singularity" means either "pole" or "essential singularity", you could start by considering the case that $f$ and $g$ have poles of (finite) orders $n$ and $m$ in $\mathbb{N}$, respectively. Next, you could look at the case that one or both of $f$ and $g$ have essential singularities (i.e. infinitely many non-zero terms occur in the principal parts of their Laurent series).