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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?

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CIT is used to prove that every holomorphic function is representable as a power series (analytic): – Ayman Hourieh Jan 31 '13 at 0:05
up vote 1 down vote accepted

(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.

(2) In fundamental theorem of calculus you assume that function has a primitive function. But in Cauchy's integral theorem you have no idea about existance of primitive function.

In CIT you really need simply connected region. There is strong geometrical connection between existence of primitive function and the region. (

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