[proofplan]
We identify the complex plane with $\mathbb{R}^2$ and write the [holomorphic function](/page/Holomorphic%20Function) as $f=u+iv$, so the real Jacobian is expressed in terms of the first partial derivatives of $u$ and $v$. The [Cauchy-Riemann equations](/page/Cauchy-Riemann%20Equations) convert this Jacobian into the real matrix of multiplication by the complex number $f'(w)$. Its determinant is therefore the squared modulus $|f'(w)|^2$, which is nonzero exactly when $f'(w)\ne 0$. Finally, we use the definition of conformality at a point for holomorphic maps: $f$ is conformal at $w$ exactly when $f'(w)\ne 0$.
[/proofplan]
[step:Write the real Jacobian using the real and imaginary parts of $f$]
Choose an [open set](/page/Open%20Set) $N \subset U$ such that $w \in N$ and $f|_N$ is holomorphic. Write $w=x_0+iy_0$ with $x_0,y_0 \in \mathbb{R}$. Define real-valued functions $u: N_{\mathbb{R}} \to \mathbb{R}$ and $v: N_{\mathbb{R}} \to \mathbb{R}$ by
\begin{align*}
f(x+iy)=u(x,y)+iv(x,y)
\end{align*}
for $(x,y) \in N_{\mathbb{R}}$, where $N_{\mathbb{R}}:=\{(x,y)\in\mathbb{R}^2:x+iy\in N\}$.
Since $f$ is holomorphic on $N$, the functions $u$ and $v$ are real differentiable at $(x_0,y_0)$, and the real Jacobian matrix $Jf_w$ is the Jacobian matrix of the map $F:N_{\mathbb{R}}\to\mathbb{R}^2$ defined by $F(x,y)=(u(x,y),v(x,y))$. Thus its entries are
\begin{align*}
(Jf_w)_{11}=\partial_x u(x_0,y_0), \quad (Jf_w)_{12}=\partial_y u(x_0,y_0), \quad (Jf_w)_{21}=\partial_x v(x_0,y_0), \quad (Jf_w)_{22}=\partial_y v(x_0,y_0).
\end{align*}
[guided]
The real Jacobian is not a complex derivative; it is the ordinary Jacobian matrix after viewing $\mathbb{C}$ as $\mathbb{R}^2$. To make this precise, we first choose an open neighbourhood $N \subset U$ of $w$ on which $f$ is holomorphic. Writing $w=x_0+iy_0$, we introduce the real coordinate domain
\begin{align*}
N_{\mathbb{R}}:=\{(x,y)\in\mathbb{R}^2:x+iy\in N\}.
\end{align*}
Now define $u:N_{\mathbb{R}}\to\mathbb{R}$ and $v:N_{\mathbb{R}}\to\mathbb{R}$ by decomposing $f$ into real and imaginary parts:
\begin{align*}
f(x+iy)=u(x,y)+iv(x,y).
\end{align*}
Under the identification $x+iy\leftrightarrow (x,y)$, the complex-valued map $f$ becomes the real map $F:N_{\mathbb{R}}\to\mathbb{R}^2$ given by $F(x,y)=(u(x,y),v(x,y))$. Because $f$ is holomorphic near $w$, this real map is differentiable at $(x_0,y_0)$. Therefore the real Jacobian matrix $Jf_w$ is obtained by differentiating the two real component functions $u$ and $v$ with respect to the real coordinates $x$ and $y$:
\begin{align*}
(Jf_w)_{11}=\partial_x u(x_0,y_0), \quad (Jf_w)_{12}=\partial_y u(x_0,y_0), \quad (Jf_w)_{21}=\partial_x v(x_0,y_0), \quad (Jf_w)_{22}=\partial_y v(x_0,y_0).
\end{align*}
This is the coordinate bridge between [complex differentiability](/page/Complex%20Differentiability) and the real matrix whose invertibility appears in the theorem.
[/guided]
[/step]
[step:Use the Cauchy-Riemann equations to identify the entries of $Jf_w$]
Since $f$ is holomorphic at $w$, the Cauchy-Riemann equations hold at $(x_0,y_0)$:
\begin{align*}
\partial_x u(x_0,y_0)=\partial_y v(x_0,y_0), \quad \partial_y u(x_0,y_0)=-\partial_x v(x_0,y_0).
\end{align*}
Moreover, the complex derivative is
\begin{align*}
f'(w)=\partial_x u(x_0,y_0)+i\,\partial_x v(x_0,y_0).
\end{align*}
Thus
\begin{align*}
\operatorname{Re} f'(w)=\partial_x u(x_0,y_0), \quad \operatorname{Im} f'(w)=\partial_x v(x_0,y_0).
\end{align*}
Substituting these identities into the entries of $Jf_w$ gives
\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*}
[/step]
[step:Compute the determinant of the real Jacobian]
Let $a:=\operatorname{Re} f'(w)$ and $b:=\operatorname{Im} f'(w)$. From the preceding step,
\begin{align*}
(Jf_w)_{11}=a, \quad (Jf_w)_{12}=-b, \quad (Jf_w)_{21}=b, \quad (Jf_w)_{22}=a.
\end{align*}
Using the determinant formula for a real $2\times 2$ matrix, we obtain
\begin{align*}
\det Jf_w=(Jf_w)_{11}(Jf_w)_{22}-(Jf_w)_{12}(Jf_w)_{21}.
\end{align*}
Substituting the four entries gives
\begin{align*}
\det Jf_w=a^2-(-b)b=a^2+b^2.
\end{align*}
Since $a=\operatorname{Re} f'(w)$ and $b=\operatorname{Im} f'(w)$, this is exactly
\begin{align*}
\det Jf_w=|f'(w)|^2.
\end{align*}
[/step]
[step:Relate invertibility of $Jf_w$ to conformality at $w$]
For a real $2\times 2$ matrix, invertibility is equivalent to having nonzero determinant. Therefore
\begin{align*}
Jf_w \text{ is invertible} \iff \det Jf_w \ne 0.
\end{align*}
Using the determinant identity just proved,
\begin{align*}
\det Jf_w \ne 0 \iff |f'(w)|^2 \ne 0 \iff f'(w)\ne 0.
\end{align*}
By the definition of conformality at a point for a holomorphic map, $f$ is conformal at $w$ exactly when $f'(w)\ne 0$. Hence $f$ is conformal at $w$ if and only if $Jf_w$ is invertible. In this case $f'(w)\ne 0$, so
\begin{align*}
\det Jf_w=|f'(w)|^2>0.
\end{align*}
This proves all asserted equivalences and the displayed formula for the determinant.
[/step]
