Complex Analysis

A function $f: \mathbf{\Omega}\subset\mathbb{C}\to\mathbb{C}$ is called holomorphic (or analytic) if

This is more familiar as the Cauchy-Riemann conditions when expressed in terms of its components z = x + iy and f(z) = u + iv, namely $$ \Large{ \begin{align} \frac{\partial f}{\partial x} + i\,\frac{\partial f}{\partial y} &= 2\,\frac{\partial\,f}{\partial\,\bar{z}}\\ \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} &= 0\\ \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} &= 0 \end{align}} $$

When integrating over closed, simply connected curve $\gamma$