Cauchy-Riemann Equations

Definition

A complex function $f(z)$ can be seen as a function of two real variables. Hence
$f(z)=f(x+iy)=f(x,y).$
When interpreted appropriately.

For instance

$f(z)=z^2=(x+iy{)}^2=x^2-y^2+2ixy$

Suppose $f$ is differentiable, we should expect relations between $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$.