Logarithm of Infinite Product of Complex Functions

Theorem
Let $X$ be a weakly locally compact connected topological space.

Let $\left \langle {f_n} \right \rangle$ be a sequence of nonzero continuous mappings $f_n: X \to \C$.

Then the following are equivalent:
 * $(1): \quad$ The product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converges locally uniformly to $f$.
 * $(2): \quad$ The series $\displaystyle \sum_{n \mathop = 1}^\infty \log f_n$ converges locally uniformly to $\log f + 2k\pi i$ for some integer $k\in\Z$.

1 implies 2
It suffices to show that:
 * $\displaystyle \sum_{n \mathop = 1}^\infty \Re(\log f_n)=\Re(\log z)$ locally uniformly
 * $\displaystyle \sum_{n \mathop = 1}^\infty \Im(\log f_n)=\Im(\log z) + 2k\pi$ locally uniformly for some $k\in\Z$

Let $K\subset X$ be compact.

2 implies 1
Follows from Complex Exponential is Uniformly Continuous on Half-Planes.

Also see

 * Logarithm of Infinite Product of Complex Numbers