# Differences between Cauchy integral theorem and fundamental theorem for integral calculus over a cycle

There is a theorem (a complex analogous to the fundamental theorem of calculus) that states that if $f$ is a continuous function having a primitive in a region of the complex plane containing a rectifiable curve $\gamma$, then $\int_{\gamma} f$ depends only on the endpoints of $\gamma$. So, if $\gamma$ is a rectifiable cycle one has $$\int_{\gamma} f = 0.$$ I would like to know what is the difference with Cauchy's integral theorem (CIT): in CIT we have a differentiable $f$ over a simply connected region with a rectifiable cycle $\gamma$ in it. Then $\int_{\gamma} = 0$. My questions are

(1) If $f$ is differentiable (and so, analytic) in a subset of $\mathbb{C}$, does $f$ have a primitive in that subset? If so, if that subset contains a rectifiable cycle, is CIT a fundamental's theorem consequence?

(2) what are the main differences between the two theorems?

-
CIT is used to prove that every holomorphic function is representable as a power series (analytic): en.wikipedia.org/wiki/Analyticity_of_holomorphic_functions – Ayman Hourieh Jan 31 '13 at 0:05

(1) No it does not have to have a primitive. Take $\frac{1}{x}$ on $\mathbb{R}\setminus \{0\}$, it does not have a primitive function.