The discussion focuses on the behavior of analytic functions with singularities at z=0, specifically examining whether the products f(z)^2 or f(z)g(z) also exhibit singularities at the same point. Participants suggest using Laurent series to analyze the coefficients in the product of these series, which provides insight into the nature of the singularities. The conversation emphasizes that if f and g have poles of finite orders n and m, their products will also have singularities, while essential singularities require a deeper examination of the series terms.
PREREQUISITESMathematicians, students of complex analysis, and anyone interested in the behavior of analytic functions and their singularities.
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?ludwig said: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?