Cauchy-Riemann Equations

Let $f(x+iy)=u(x,y)+iv(x,y)$ . The Cauchy-Riemann equations are:

$\begin{aligned} u_x&=v_y\\ u_y&=-v_x. \end{aligned}$

Alternative Form (Wirtinger Derivative)

The Cauchy-Riemann equations can be written as a single equation $\displaystyle \frac{\partial f}{\partial\bar z}=0$ where $\displaystyle \frac{\partial}{\partial\bar z}=\frac 12(\frac{\partial}{\partial x}+i\frac{\partial}{\partial y})$ is the Wirtinger derivative with respect to the conjugate variable.

Goursat’s Theorem

Suppose $f=u+iv$ is a complex-valued function which is differentiable as a function $f:\mathbb{R}^2\to\mathbb{R}^2$ . Then $f$ is analytic in an open complex domain $\Omega$ iff it satisfies the Cauchy-Riemann equations in the domain.