Derivative of Infinite Product

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Theorem

Complex Analytic Functions

Let $D \subset \C$ be open.

Let $\left\langle{f_n}\right\rangle$ be a sequence of analytic functions $f_n: D \to \C$.

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


Then:

$\displaystyle f' = \sum_{n \mathop = 1}^\infty f_n'\cdot \prod_{\substack{k \mathop = 1 \\ k\mathop \ne n} }^\infty f_k$

and the series converges locally uniformly in $D$.


Also see