Given any sequence $(a_n)_{n \in \mathbf{N}}$ is $\sum_{n \geq 0} a_n e^{2 \pi i n z}$ holomorphic on the upper half plane?

Question

I've seen quite often that people consider some arbitrary sequence $(a_n)_{n \in \mathbf{N}}$ (say of real numbers), and form the sum $\sum_{n \geq 0} a_n e^{2 \pi i n z}$, $z \in \mathbf{H}$. Usually this function is then treated like a holomorphic function, without further checking this. Is this always true? If not, what are the conditions that ensure holomorphicity?

Thanks!

Answer 1

I hope you've never seen people talk about that sum for an arbitrary sequence $(a_n)$. More likely you've seen them talk about it for sequences such that the author considered convergence to be obvious.

If you say $w=e^{2\pi i z}$ then $|w|<1$ for $z$ in the upper half plane, and any $w$ with $|w|<1$ arises this way. So your sum converges (hence converges uniformly on compact sets, hence converges to a holomorphic function) if and only if the series $\sum a_n w^n$ has radius of convergence at least $1$.