Path independence

A contour integral whose value depends only on the endpoints, not on which contour joins them.

A contour integral of $f$ is path-independent on a domain $D$ if $\int_C f(z)\,dz$ has the same value for every contour $C$ in $D$ joining a given start point to a given end point, regardless of the route taken between them. Path independence on $D$ holds exactly when $f$ has an antiderivative throughout $D$ , and is equivalent to $\int_C f(z)\,dz = 0$ around every closed contour $C$ in $D$ .

Examples

Since $f(z)=z$ has antiderivative $z^2/2$ everywhere, $\int_C z\,dz$ from $0$ to $i$ gives $-\tfrac12$ along any contour at all.
$\int_C \frac1z\,dz$ around a closed contour encircling the origin equals $2\pi i$ , not $0$ , exactly because $1/z$ has no antiderivative on a domain containing that closed loop.