# Find all entire functions having a root

EDIT: I am asking just by curiosity, this is not an assignment. Usually I think hard about my mathematical doubts, but this time I am (both) curious and lazy.

Determine all the sequences $(a_n)_{n\geq0}$ in $\mathbb C$ such that the function $f$ given by $f(z)=\sum_{n=0}^\infty a_nz^n$ is entire and takes the value $0$.

The fundamental theorem of algebra guarantees that sequences $(a_n)$ with $a_n=0$ for $n$ sufficiently large satisfy the required conditions. I would like to know if all the other sequences have already determined elsewhere.

• Is this an assignment or have you thought this question out? (this is useful to know because if it is not an assignment, it could be tremendously difficult to answer) – user384138 Jan 7 '17 at 15:41

1. Picards theorem says that any non-constant entire function takes all values in $\mathbb{C}$ except possibly a single value. This already tell us than almost all entire functions have a zero.
2. The entire functions that do not have a zero can be written (see also Weierstrass factorization theorem) on the form $f(z) = e^{g(z)}$ for some entire function $g$.
3. $f$ is entire if and only if the sequence $\{a_n\}_{n=0}^\infty$ satisfy $\lim_{n\to\infty}|a_n|^{1/n} = 0$.
Based on these facts we have that almost all sequences that satisfy $\lim_{n\to\infty}|a_n|^{1/n} = 0$ will produce an entire function that has a zero. Those that do not are most easily described by their functional form rather than by the sequence $a_n$.