Antiderivative (complex sense)

A function $F$ with $F'(z)=f(z)$ throughout a domain -- lets a contour integral be evaluated from endpoints alone.

An antiderivative of $f$ on a domain $D$ is a function $F$ , analytic on $D$ , with $F'(z)=f(z)$ at every point of $D$ . When such an $F$ exists, the fundamental theorem for contour integrals turns $\int_C f(z)\,dz$ into $F(z_1)-F(z_0)$ for any contour $C$ in $D$ from $z_0$ to $z_1$ -- exactly the real-variable fundamental theorem of calculus, carried over to the complex setting -- which is what makes finding an antiderivative so valuable whenever one exists.

Examples

$F(z)=z^2/2$ is an antiderivative of $f(z)=z$ on all of $\mathbb C$ , since $F'(z)=z$ .
$f(z)=1/z$ has an antiderivative on any simply connected domain avoiding the origin, but none on the full punctured plane $\mathbb C\setminus\{0\}$ .