Let $U \subset \mathbb{C}$ be open, let $w \in U$, and let $f: U \to \mathbb{C}$ be holomorphic on an open neighbourhood of $w$ contained in $U$. Identify $\mathbb{C}$ with $\mathbb{R}^2$ by $x+iy \mapsto (x,y)$, and let $Jf_w \in \mathbb{R}^{2 \times 2}$ denote the real Jacobian matrix of $f$ at $w$ under this identification. Then $f$ is conformal at $w$ if and only if $Jf_w$ is invertible.

Moreover, writing $w=x_0+iy_0$, the entries of $Jf_w$ are

\begin{align*} (Jf_w)_{11}=\operatorname{Re} f'(w), \quad (Jf_w)_{12}=-\operatorname{Im} f'(w), \quad (Jf_w)_{21}=\operatorname{Im} f'(w), \quad (Jf_w)_{22}=\operatorname{Re} f'(w). \end{align*}

Consequently,

\begin{align*} \det Jf_w = |f'(w)|^2, \end{align*}

and in the conformal case this determinant is strictly positive.