Proving Cauchy Riemann Equations for General Functions

Question

How would one prove that Cauchy-Riemann equations hold for any holomorphic functions? I used the definition of complex differentiability to achieve the second equation ( $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial y}$) by approaching the limit from different sides, but I cant seem to understand where the first equation($ \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$) comes from.

Answer 1

Note that the definition of the complex differentiability of $f$ at $a$ implies the existence of a $\phi : \mathbb{C} \to \mathbb{C}$ continuous at $a$ such that $\phi(a) = f'(a)$ and \begin{align*} f(z) - f(a) = \phi(z)(z - a). \end{align*} Treating $\mathbb{C}$ as $\mathbb{R}^2$, $\phi(a)$ is a linear map from $\mathbb{R}^2$ to $\mathbb{R}^2$. Thus $f$ is totally differentiable at $a$, and there exists a Jacobian $J$ such that $J(f;a)(z) = wz$ for some complex number $w$.

In other words, the local effect of a complex differentiable function is multiplication by a complex number (a rotation and a dilation).

The result follows quickly: since the linear approximation of a complex differentiable function $f = u(x, y) + v(x, y)i$ at $x + yi = z \in \mathbb{C}$ is given by the Jacobian \begin{align*} \begin{pmatrix} \frac{\partial u}{\partial x}(z) & \frac{\partial u}{\partial y}(z)\\ \frac{\partial v}{\partial x}(z) & \frac{\partial v}{\partial y}(z) \end{pmatrix}, \end{align*} and transformations that represent a rotation and a dilation are of the form \begin{align*} \begin{pmatrix} \alpha & -\beta \\ \beta & \alpha \end{pmatrix}, \end{align*} we obtain \begin{align*} \frac{\partial u}{\partial x}(z) = \frac{\partial v}{\partial y}(z) \quad \text{and} \quad \frac{\partial u}{\partial y}(z) = -\frac{\partial v}{\partial x}(z). \end{align*} Hope this helps.