What is the common way of showing that infinite series is holomorphic?

Question

I saw frequently in some literature that "function is absolutely convergent and uniformly bounded so it is holomorphic" but I don't understand how this follows. Generally in number theory, how does one show that infinite series is holomorphic?

The only theorem I know is the following one

If $\{f_n\}^\infty_{n=1}$ is a sequence of holomorphic functions that converges uniformly to a function $f$ in every compact subset of $\Omega$, then $f$ is holomorphic in $\Omega$.

Also, I have Morera's Theorem, for which I'm not quite sure how I should use in practice.

Would someone give me an insight or point out a relevant theorem?

Answer 1

I'm not sure whether you typed your sentence correctly, or not, but here's my attempt at a best guess regarding what you might have meant.

Morera's Theorem is good start. Now suppose you have a sequence of holomorphic functions $\{g_n\}_{n=1}^{\infty}$ all defined on some open set $\Omega\subset \Bbb{C}$.

For each $k$, let $f_k=g_1+\dots +g_k$ be the $k^{th}$ partial sum. If you can show somehow that there is an $f:\Omega\to\Bbb{C}$ such that $f_k\to f$ uniformly on compact subsets of $\Omega$, then by the theorem you mention, it follows that $f=\sum_{n=1}^{\infty}g_n$ is holomorphic.

Now, how do we show that the partial sums of a sequence of functions converges uniformly? Well, the simplest and most classical test is the Weierstrass M-test, which in great generality says the following:

Let $A$ be any non-empty set (doesn't even have to be a subset of $\Bbb{C}$), and let $V$ be a Banach space (eg $\Bbb{C}$). Given a sequence of functions $\{g_n:A\to V\}_{n=1}^{\infty}$, let $M_n:=\sup\limits_{z\in A}|g_n(z)|_V$. If $\sum_{n=1}^{\infty}M_n<\infty$ then the partial sums $\left\{\sum_{k=1}^ng_k\right\}_{n=1}^{\infty}$ converge absolutely and uniformly.

In short, this says "normal convegrence/convergence in supremum norm implies absolute and uniform convergence". So, if you combine the Weirstrass M-test and that theorem you know and apply it to each compact subset of $\Omega$, it gives you a simple way of checking that a series of holomorphic functions is again holomorphic.