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:

$(g\circ f{)}^{\prime }(z_0)=g^{\prime }(f(z_0))f^{\prime }(z_0)$

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:

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