Primitive

A primitive of \(f\) on \(D\) is an analytic function \(F: D \mapsto \mathbb{C}\) such that \(F' = f\) on \(D\), where \(D \subset \mathbb{C}\).

If \(f\) is continuous on a domain \(D\) and if \(f\) has a primitive \(F\) in \(D\), then for any curve \(\gamma: [a,b] \mapsto D\),

\[ \begin{aligned} \int_\gamma f(z)\,dz = F(\gamma(b)) - F(\gamma(a)) \\ \end{aligned} \]

Observe that

• The integral depends only on the initial and terminal points of \(\gamma\) !
• The critical assumption of \(f\) having a primitive in \(d\).

When does \(f\) have a primitive ?

By Goursat Theorem, if \(D\) is a simply connected domain in \(\mathbb{C}\), and \(f\) is analytic in \(D\), then \(f\) has a primitive in \(D\).

More at Analysis of a Complex Kind.