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Fundamental theorem of line integral states that for any function $f$ that has an antiderivative $F$, integrating $f$ from point $a$ to point $b$ yields $F(b) - F(a)$, which would imply integration over a closed path yields $0$; However, Cauchy theorem requires the function to be analytic to guarantee $0$ on closed path integration. So does this mean any function that has primitive function $F$ will automatically be analytic and vice versa?

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    $\begingroup$ But if by primitive of $f$ you mean there is an $F$ with$F'=f$, then $F$ itself is differentiable and therefore analytic -- and then so is $f$. $\endgroup$
    – gary
    Jul 26, 2017 at 17:40
  • $\begingroup$ Are you familiar with Morera's theorem? $\endgroup$
    – Chappers
    Jul 26, 2017 at 17:42
  • $\begingroup$ @gary Thank you for the answer. But what about the reverse? If f is analytic do we always have antiderivative such that $F'=f$? $\endgroup$
    – Daniel Li
    Jul 26, 2017 at 17:45
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    $\begingroup$ @Daniel Li. Yes, if the domain is simply-connected. $\endgroup$
    – gary
    Jul 26, 2017 at 17:49
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    $\begingroup$ @DanielLi: Please see the "Sufficiency" part of en.wikipedia.org/wiki/Antiderivative_(complex_analysis) , near the bottom, and let me know if you still have questions. $\endgroup$
    – gary
    Jul 26, 2017 at 18:04

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Yes, if $f$ is analytic and defined in a simply-connected domain $D$, it will have an analytic antiderivative given by $ F(z)=\int_{\gamma_z}f(z)dz$ , where $\gamma$ is any curve living in $D$, i.e., $\frac {d}{dz}F(z):=\int_{\gamma_z}f(t)dt=f(z)$. See, e.g.:http://planetmath.org/antiderivativeofcomplexfunction

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    $\begingroup$ note, not necessarily true if the domain is not simply connected, for example 1/z on all C $\endgroup$
    – fdzsfhaS
    Dec 30, 2018 at 3:38
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Because the derivative of a holomorphic function is itself holomorphic, any function that admits a primitive is holomorphic. However, the function $z\in \mathbb C\smallsetminus 0 \longmapsto z^{-1}\in \mathbb C$ admits no primitive on all its domain, because it winds once around the origin, and any function that admits a primitive must wind zero times around it.

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