If a function is holomorphic in a certain domain, does it mean it necessarily has an antiderivativeSay I have a holomrphic function f(z) in domain D, say | z | &gt; 4 and my function is not holomorphic for | z | ≤ 4 - it has some poles there. Does it mean there is necessarily an antiderivative for f(z)?I know I can use Morera's theorem and calculate all the residues inside | z | ≤ 4 and check if their sum is zero. However what seemed more intuitive to me is to prove that since ∫ γ f ( z ) d z for every γ contained in D, depends only on the start and end points, there exists an antiderivative for f(z).Is my prof correct? Can I conclude that in general, a holomorphic function in a domain D not only has infinitely many derivatives in D, but it has also infinitely many antiderivatives there?

Question

If a function is holomorphic in a certain domain, does it mean it necessarily has an antiderivativeSay I have a holomrphic function f(z) in domain D, say       |    z      |    &amp;gt;  4 and my function is not holomorphic for       |    z      |    ≤  4 - it has some poles there. Does it mean there is necessarily an antiderivative for f(z)?I know I can use Morera&#039;s theorem and calculate all the residues inside       |    z      |    ≤  4 and check if their sum is zero. However what seemed more intuitive to me is to prove that since       ∫          γ        f  (  z  )  d  z for every       γ   contained in D, depends only on the start and end points, there exists an antiderivative for f(z).Is my prof correct? Can I conclude that in general, a holomorphic function in a domain D not only has infinitely many derivatives in D, but it has also infinitely many antiderivatives there?

Jayvion Mclaughlin · Accepted Answer

Explanation:No. If   f  (  z  )  =      1    z   (on   {  z  ∈      C    ∣      |    z      |    &amp;gt;  4  }), then f has no antiderivative.But if f has the additional property that       ∫    γ    f  (  z  )        d    z dpends only upon the initional and the end point of   γ, then, yes, f has an antiderivative. It does not follow, however, that such an antiderivative will have an antiderivative as well. Take       1          z      2      , for instance.