Chain rule for complex differentiation

Definition

Complex-differentiable at a point

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

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