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 $\left \langle {f_n} \right \rangle$ 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:

  • $f_n(x)\neq0$ for $n\geq n_0$ and $x\in U$
  • The series $\displaystyle \sum_{n \mathop = n_0}^\infty \log f_n$ converges uniformly on $U$ to $\log g + 2k\pi i$, where $g = \displaystyle \prod_{n\mathop =n_0}^\infty f_n$.


Outline of Proof

We construct a neighborhood of $x_0$ on which $\displaystyle \sum_{n \mathop = n_0}^\infty \log f_n$ and $\log g$ 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 $\displaystyle \prod_{n \mathop = N}^\infty f_n(x)\notin\R^-$ for all $x\in K$.

By Factors in Uniformly Convergent Product Converge Uniformly to One, there exists $M\in\N$ such that $|f_n(x) - 1|\leq\frac12$ for $n\geq M$ and $x\in K$.

Let $n_0 = \max(N,M)$.

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 $\displaystyle \sum_{n \mathop = n_0}^\infty \log f_n = \log g + 2k\pi i$ 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, $\log g$ is continuous on $K$.

By the Heine-Cantor Theorem, $\log$ is uniformly continuous on $\overline B(1,\frac12)$.

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

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

Because $\log g$ and $\displaystyle \sum_{n \mathop = n_0}^\infty \log f_n$ 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 $U$.

$\blacksquare$