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 $f : U \to C$ with $U \subseteq C$ is a complex function. We can write $f$ in terms of the component functions

f (x + i y) = u (x, y) + i v (x, y)

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

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

The derivative of $g$ is given by the Jacobian matrix, which in this case is

J g = [\begin{matrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{matrix}] .

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 $J g$ to be like a rotation and a stretch. Given that a rotation matrix looks like

[\begin{matrix} \cos (θ) & - \sin (θ) \\ \sin (θ) & \cos (θ) \end{matrix}]

we would expect

J g = [\begin{matrix} \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} \\ \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{matrix}] = r [\begin{matrix} \cos (θ) & - \sin (θ) \\ \sin (θ) & \cos (θ) \end{matrix}]

for some $r$ and $θ$ in order for $J g$ to represent complex multiplication.

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

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

and therefore we have

a = \frac{\partial u}{\partial x} = \frac{\partial v}{\partial y} and b = \frac{\partial v}{\partial x} = - \frac{\partial u}{\partial y}

which are exactly the Cauchy-Riemann equations.

Note that with arbitrary $r$ and $θ$ 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{matrix} a & - b \\ b & a \end{matrix}]

for some $a, b \in R$ .