Complex Differentiability: Sums, Differences, Products, and Quotients

Complex Differentiability of Sums, Differences, Products, and Quotients of Complex Differentiable Functions

Theorem 1: Let $f, g : A \to \mathbb{C}$ be a complex function and let $k \in \mathbb{C}$. If $f$ and $g$ are both complex differentiable at $z_0 \in A$ then: a) $f + g$ is complex differentiable at $z_0$ and $(f + g)'(z_0) = f'(z_0) + g'(z_0)$. b) $kf$ is complex differentiable at $z_0$ and $(kf)'(z_0) = kf'(z_0)$. c) $fg$ is complex differentiable at $z_0$ and $(fg)'(z_0) = f(z_0)g'(z_0) + f'(z_0)g(z_0)$. d) $\displaystyle{\frac{f}{g}}$ is complex differentiable at $z_0$ and $\displaystyle{\left ( \frac{f}{g} \right )'(z_0) = \frac{f'(z_0)g(z_0) - f(z_0)g'(z_0)}{[g(z_0)]^2}}$ (provided that $g'(z_0) \neq 0$).

The proofs of the result from Theorem 1 are analogous to those for real differentiable functions. We will state and prove a very similar result on the Analyticity of Sums, Differences, Products, and Quotients of Analytic Functions page later.