Relationship Between Jacobian and Complex Derivative

Considering a complex function as a function of two real variables provides a useful demonstration of

why complex differentiation is a stronger condition than differentiation in the two real variable case
where the Cauchy-Riemann equations come from
how complex derivatives "look" in 2D space

which we present below.

Suppose \(f : U \to \mathbb{C}\) with \(U \subseteq \mathbb{C}\) is a complex function. We can write \(f\) in terms of the component functions

\[ f(x + iy) = u(x, y) + iv(x, y)\]

and hence construct an analogue \(g : V \to \mathbb{R}^2\) with \(V \subseteq \mathbb{R}^2\)

\[ g\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} u(x, y) \\ v(x, y) \end{bmatrix}.\]

The derivative of \(g\) is given by the Jacobian matrix, which in this case is

\[Jg = \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \\ \end{bmatrix}.\]

The complex derivative gives a local approximation of the function in the form of complex multiplication, which is a rotation and stretch. As such, we would expect this matrix \(Jg\) to be like a rotation and a stretch. Given that a rotation matrix looks like

\[\begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}\]

we would expect

\[Jg = \begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \\ \end{bmatrix} = r \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix}\]

for some \(r\) and \(\theta\) in order for \(Jg\) to represent complex multiplication.

Some manipulation and the substitution \(a = r \cos(\theta)\) and \(b = r \sin(\theta)\) reveals

\[\begin{bmatrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \\ \end{bmatrix} = \begin{bmatrix} r \cos(\theta) & -r \sin(\theta) \\ r \sin(\theta) & r \cos(\theta) \\ \end{bmatrix} = \begin{bmatrix} a & -b \\ b & a \\ \end{bmatrix}\]

and therefore we have

\[ a = \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} \quad \text{and} \quad b = \frac{\partial v}{\partial x} = - \frac{\partial u}{\partial y}\]

which are exactly the Cauchy-Riemann equations.

Note that with arbitrary \(r\) and \(\theta\) we still have arbitrary \(a\) and \(b\) (for the same reason polar coordinates "work").

So complex differentiable functions correspond with real two-variable differentiable functions whose Jacobian is of the form

\[\begin{bmatrix} a & -b \\ b & a \\ \end{bmatrix}\]

for some \(a, b \in \mathbb{R}\).