If a function is holomorphic in a certain domain, does it mean it necessarily has...

vortoca

vortoca

Answered

2022-06-30

If a function is holomorphic in a certain domain, does it mean it necessarily has an antiderivative
Say I have a holomrphic function f(z) in domain D, say | z | > 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?

Answer & Explanation

Jayvion Mclaughlin

Jayvion Mclaughlin

Expert

2022-07-01Added 14 answers

Explanation:
No. If f ( z ) = 1 z (on { z C | z | > 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.

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