Chain rule for complex differentiation

Definition

Complex-differentiable at a point

Suppose $U, V \subset C$ are open subsets and $f : U \to C, g : V \to C$ are functions with the property that $f (U) \subset V$ . Then, we can define a function $g \circ f : U \to C$ by:

$(g \circ f) (z) = g (f (z))$

Suppose $z_{0} \in U$ is a point such that $f$ is complex-differentiable at $z_{0}$ and $g$ is complex-differentiable at $f (z_{0})$ . Then, $g \circ f$ is complex-differentiable at $z_{0}$ , and:

$(g \circ f)^{'} (z_{0}) = g^{'} (f (z_{0})) f^{'} (z_{0})$

For holomorphic functions

Suppose $U, V \subset C$ are open subsets and $f : U \to C, g : V \to C$ are holomorphic functions with the property that $f (U) \subset V$ . Then, we can define a function $g \circ f : U \to C$ by:

$(g \circ f) (z) = g (f (z))$

Then, $g \circ f$ is also a holomorphic function and for any point $z \in U$ , we have:

$(g \circ f)^{'} (z) = g^{'} (f (z)) f^{'} (z)$