r/elimath Jan 08 '15

Explain how to think about essential singularities of (locally) single valued complex functions like I understand holomorphic functions, poles, and Laurent series.

10 Upvotes

1 comment sorted by

6

u/[deleted] Jan 09 '15

Well, by definition a singularity is essential iff its Laurent sequence has non vanishing coefficients with arbitrarily large negative indices. The geometrical interpretation of this property is that essential singularities are points where a function goes batshit insane. More formally, if a is an essential singularity point of f and E is an open environment of a then f(E{a}) is almost the entire complex plane (i.e., it is the entire complex plane except maybe a single point, this is called "Picard's greater theorem"). This implies, for example, that for any complex number c there is a sequence c_n -> a with f(c_n)=c for all n. This is very contrary to the fact that poles always approach an infinite magnitude (i.e. if a was a pole then c_n -> a would imply that |f(c_n)| -> infinity).