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.

$\blacksquare$

Also see

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