The geometry of complex differentiability (9/6/2020)

Complex differentiability is usually viewed as an analytic property of functions. But this perspective suggests we might look for a geometric characterization. Take a function $f(z)=u(z)+iv(z)$ with real-valued $u,v$. Now we can define the partial derivatives of $u$ and $v$ with respect to $x=\Re(z)$ and $y=\Im(z)$.

Where $f$ is complex differentiable, it is straightforward to derive the celebrated Cauchy-Riemann equations $u_x=v_y$ and $u_y=-v_u$. We can use these relations to manipulate the complex differential

$$df:=\begin{pmatrix}u_x & u_y \\ v_x & v_y\end{pmatrix}\stackrel{CauRie}{=}\begin{pmatrix}u_x & u_y \\ -u_y & u_x\end{pmatrix}\stackrel{LinAlg}{=}u_x\begin{pmatrix}1 & 0 \\ 0 & 1\end{pmatrix}+u_y\begin{pmatrix}0 & 1 \\ -1 & 0\end{pmatrix}.$$

Thus the pushforward of a vector in the tangent space to a complex differentiable point of $f$ is the weighted sum of a scaling and a rotation by $\pi/2$. So the differential is a conformal map — angles are preserved.

It is easy to see that this characterization is also sufficient. For if $df$ is the weighted sum of a scaling and a rotation by $\pi/2$, we can derive the Cauchy-Riemann equations by working in reverse, and we’re done.

This post results from a discussion I had with Eyal Markman following a lecture on complex analysis.