Derivative of Infinite Product

Theorem

Complex Analytic Functions

Let $D \subset \C$ be open.

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

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

Then:

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

and the series converges locally uniformly in $D$.

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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

• Derivative of Product
• Logarithmic Derivative of Infinite Product