Combination Theorem for Complex Derivatives/Combined Sum Rule

Theorem

Let $D$ be an open subset of the set of complex numbers.

Let $f, g: D \to \C$ be complex-differentiable functions on $D$.

Let $c, w \in \C$.

Let $\map {\dfrac \d {\d z} } {c f + w g}$ denote the derivative of $c f + w g$.

Then:

$\map {\map {\dfrac \d {\d z} } {c f + w g} } z = c \dfrac \d {\d z} \map f z + w \dfrac \d {\d z} \map g z$

for all $z \in D$.

Proof

\(\ds c \dfrac \d {\d z} \map f z + w \dfrac \d {\d z} \map g z\)

\(=\)

\(\ds \map {\map {\dfrac \d {\d z} } {c f} } z + \map {\map {\dfrac \d {\d z} } {w g} } z\)

Combination Theorem for Complex-Differentiable Functions: Multiple Rule

\(\ds \)

\(=\)

\(\ds \map {\map {\dfrac \d {\d z} } {c f + w g} } z\)

Combination Theorem for Complex-Differentiable Functions:Sum Rule

$\blacksquare$

Sources

• 2001: Christian Berg: Kompleks funktionsteori $\S 1.1$