Logarithmic Derivative of Product of Analytic Functions

Theorem
Let $D \subset \C$ be open.

Let $f, g: D \to \C$ be analytic.

Let $f g$ be their pointwise product.

Let $z \in D$ with $\map f z \ne 0 \ne \map g z$.

Then:
 * $\dfrac {\map {\paren {f g}' } z} {\map {\paren {f g} } z} = \dfrac{\map {f'} z} {\map f z} + \dfrac {\map {g'} z} {\map g z}$

Proof
Follows directly from Product Rule for Complex Derivatives.

Also see

 * Product of Analytic Functions is Analytic
 * Product Rule for Complex Derivatives
 * Logarithmic Derivative of Infinite Product of Analytic Functions