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.

# posted by rot13(Unafba Pune) @ 7:28 AM