Logarithmic Derivative of Infinite Product

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Complex Analytic Functions

Let $D\subset\C$ be open.

Let $(f_n)$ be a sequence of analytic functions $f_n:D\to\C$.

Let none of the $f_n$ be identically zero on any open subset of $D$.

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

Then $\displaystyle \frac{f'}f = \sum_{n \mathop = 1}^\infty \frac{f_n'}{f_n}$

and the series converges locally uniformly in $D\setminus\{z\in D : f(z) = 0\}$.

Also see