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 $g:=f'$ 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.