Analytic functions

A complex function $f(z)=u(x,y)+iv(x,y)$ is analytic at a point if it is complex differentiable in a neighborhood of that point. For functions with continuous first partial derivatives, this happens exactly when the Cauchy–Riemann equations hold: $\frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}$ and $\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}$. These equations link the real and imaginary parts and are the primary test for analyticity. For example, $f(z)=z^2$ is analytic everywhere because its real and imaginary parts satisfy these equations at every point.