Logarithmic Derivative of Infinite Product

Theorem

Complex Analytic Functions

Let $D \subseteq \C$ be open.

Let $\sequence {f_n}$ 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 $\ds \prod_{n \mathop = 1}^\infty f_n$ converge locally uniformly to $f$.

Then:

$\ds \dfrac {f'} f = \sum_{n \mathop = 1}^\infty \frac {f_n'} {f_n}$

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

A particular theorem is missing. In particular: the same for real functions and meromorphic functions You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding the theorem. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{TheoremWanted}} from the code.

Also see

• Logarithmic Derivative of Product
• Derivative of Infinite Product