Infinite Product of Analytic Functions is Analytic
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Theorem
Let $D \subset \C$ be an open set.
Let $\sequence {f_n}$ be a sequence of analytic functions $f_n: D \to \C$.
Let $\ds \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$.
Then $f$ is analytic.
Proof
Follows directly from:
- Partial Products of Uniformly Convergent Product Converge Uniformly
- Uniform Limit of Analytic Functions is Analytic
$\blacksquare$
Also see
Sources
- 1973: John B. Conway: Functions of One Complex Variable ... (previous) ... (next) $VII$: Compact and Convergence in the Space of Analytic Functions: $\S5$: Weierstrass Factorization Theorem: Theorem $5.9$