# Complex Exponential is Uniformly Continuous on Half-Planes/Corollary

< Complex Exponential is Uniformly Continuous on Half-Planes(Redirected from Uniform Absolute Convergence of Infinite Product of Complex Functions/Lemma)

## Corollary to Complex Exponential is Uniformly Continuous on Half-Planes

Let $X$ be a set.

Let $(g_n)$ be a family of mappings $g_n : X\to\C$.

Let $g_n$ converge uniformly to $g:X\to\C$.

Let there be a constant $a\in\R$ such that $\Re( g(x)) \leq a$ for all $x\in X$.

Then $\exp g_n$ converges uniformly to $\exp g$.

## Proof

By uniform convergence, there exists $N>0$ such that $|g_n(x)-g(x)|\leq1$ for all $n>N$.

Then $\Re(g_n(x))\leq a+1$.

The result now follows from

- Complex Exponential is Uniformly Continuous on Half-Planes, applied to the half-plane $\{z\in\C : \Re(z)\leq a+1\}$
- Uniformly Continuous Function Preserves Uniform Convergence

$\blacksquare$

## Sources

- 1973: John B. Conway:
*Functions of One Complex Variable*... (next) $VII$: Compact and Convergence in the Space of Analytic Functions: $\S5$: Weierstrass Factorization Theorem: Lemma $5.7$