Consider a complex valued function \(f\colon U\to \mathbb{C}\). We denote its real part as \(u\) and its imaginary part as \(v\), i.e. \(f=u+iv\). A function \(f\colon U\to \mathbb{C}\) is holomorphic , if and only if its real and imaginary part suffices the Cauchy-Riemann equations:
\begin{equation*} u_x=v_y\qquad \text{and} \qquad u_y=-v_x, \end{equation*}where \(z=x+iy\).