Fundamental theorem of complex calculus

This article gives the statement, and possibly proof, of a basic fact in complex analysis.
View a complete list of basic facts in complex analysis

Statement

Suppose $f$ is a holomorphic function on a domain $U\subset \mathbb{C}$, and $g:=f^{\prime }$ is its complex differential. Suppose $\gamma$ is a piecewise smooth curve in $U$, starting at $z_0$ and ending at $z_1$.

Then, we have:

${\int }_{\gamma }g(z)dz=f(z_1)-f(z_0)$

This is the complex analog of the fundamental theorem of calculus.