Cauchy-Riemann Equations

The Cauchy-Riemann equations relate two formulas for the derivative of a Complex Functions, with their real- and imaginary parts must be equal, respectively. The derivative of the complex function must be the same, no matter from with direction the derivative is approached.

$$\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\text{and}\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$$