Logarithm of Infinite Product of Complex Functions/Corollary

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Corollary to Logarithm of Infinite Product of Complex Functions

Let $X$ be a locally compact and locally connected metric space.

Let $\sequence {f_n}$ be a sequence of continuous mappings $f_n: X \to \C$.

Let the product $\displaystyle \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$.

Let $x_0 \in X$.


Then there exist $n_0 \in \N$, $k \in \Z$ and a neighborhood $U$ of $x_0$ such that:

$(1): \quad \map {f_n} x \ne 0$ for $n \ge n_0$ and $x \in U$
$(2): \quad$ The series $\displaystyle \sum_{n \mathop = n_0}^\infty \ln f_n$ converges uniformly on $U$ to $\ln g + 2 k \pi i$, where $g = \displaystyle \prod_{n \mathop = n_0}^\infty f_n$.


Outline of Proof

We construct a neighborhood of $x_0$ on which $\displaystyle \sum_{n \mathop = n_0}^\infty \ln f_n$ and $\ln g$ are continuous, so that $k$ is continuous and thus constant.


Proof

Let $K$ be a compact neighborhood of $x_0$.

By Tail of Uniformly Convergent Product Converges Uniformly to One, there exists $N \in \N$ such that $\displaystyle \prod_{n \mathop = N}^\infty \map {f_n} x \notin \R^-$ for all $x \in K$.

By Factors in Uniformly Convergent Product Converge Uniformly to One, there exists $M \in \N$ such that $\size {\map {f_n} x - 1} \le \dfrac 1 2$ for $n \ge M$ and $x \in K$.

Let $n_0 = \map \max {N, M}$.

Let $g = \displaystyle \prod_{n \mathop = n_0}^\infty f_n$.

By Logarithm of Infinite Product of Complex Functions, there exists $k: K \to \Z$ such that $\displaystyle \sum_{n \mathop = n_0}^\infty \ln f_n = \ln g + 2 k \pi i$ uniformly on $K$.

We show that $k$ is constant on some neighborhood $U \subset K$.

By Uniform Product of Continuous Functions is Continuous and Complex Logarithm is Continuous Outside Branch, $\ln g$ is continuous on $K$.

By the Heine-Cantor Theorem, $\ln$ is uniformly continuous on $\map {\overline B} {1, \dfrac 1 2}$.

By Uniformly Continuous Function Preserves Uniform Convergence, $\displaystyle \sum_{n \mathop = n_0}^\infty \ln f_n$ converges uniformly on $K$.

By the Uniform Limit Theorem, $\displaystyle \sum_{n \mathop = n_0}^\infty \ln f_n$ is continuous.

Because $\ln g$ and $\displaystyle \sum_{n \mathop = n_0}^\infty \ln f_n$ are continuous, so is $k$.

Let $U\subset K$ be a connected neighborhood of $x_0$.

By Continuous Mapping from Connected to Discrete Space is Constant $k$ is constant on $U$.

$\blacksquare$