Is there a classification of isolated essential singularities?

Solution 1:

The problem with the quotient by multiplication by invertible functions is that it barely makes a dent in the size of the space. The quotient remains too big to admit any explicit set of representatives (in contrast to the case of poles, which get represented by $\{z^{-n}:n\ge 1\}$). For example, a function can have any number of asymptotic values, to which it tends along various complicated curves terminating at the essential singularity. This structure remains untouched by the quotient. On the other hand, taking the quotient we lose the structure of vector space.

A better way to take a quotient of singular germs is additively: up to addition of a holomorphic function in a neighborhood of $0$. Then the orbits have a nice set of representatives: namely, functions holomorphic in $\widehat{\mathbb C}\setminus \{0\}$ and vanishing at $\infty$. Indeed, any function holomorphic in $\{z:0<|z|<r\}$ can be expanded into Laurent series $\sum_{n\in\mathbb Z}c_n z^n$; we get a representative by keeping only the terms with $n<0$.

Having taken the quotient in the preceding paragraph, we can just as well replace $z$ by $1/z$, thus putting equivalence classes of essential singularities in bijection with transcendental entire functions vanishing at $0$. The "vanishing at $0$" is more of a distraction, of which we can get rid by dividing by $z$. So, the question becomes: how to classify transcendental entire functions? Given the existence of several monographs written on the subject of entire functions, we can't have much hope for an exhaustive structure theorem.

Although the Weierstrass factorization theorem allows us to write down $f$ as a certain product, this product involves $e^g$, where $g$ is just another entire function. The factorization works much better when $f$ has finite order because $g$ becomes a polynomial then (Hadamard factorization). In this way, we get a classification of entire functions of finite order, and - returning to the original formulation - of essential singularities at $0$ such that $$\limsup_{r\to 0}\frac{1}{\log (1/r)}\log\log \sup_{|z|\ge r} |f(z)|<\infty \tag1$$ Every such singularity is essentially described by a polynomial and a sequence of complex numbers converging to $0$.