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I am wondering if this is true. i did a proof but i'm not sure about it.

take $D \subseteq \mathbb C$ to be a domain and let $f: D \to \mathbb C$ be a holomorphic on $D$. then $f$ is analytic on $D$ so if $a \in D$ then we can find a disk $D(a,r)$ and a power series $\sum_{k=0}^{\infty} a_k (z - a)^k$ converging to $f(z)$ for every $z$ in $D(a,r)$. take $F(z) = \sum_{k=0}^{\infty} (c_k/(k+1)) (z-a)^{k+1}$ then $F$ is holomorphic on $D(a,r)$ and $F'(z) = f(z)$ for every $z \in D(a,r)$ so in particular $F'(a) = f(a)$.

So if I define $F$ on $D$ such that $F(z)$ is equal to that series over the corresponding disk around each point in $D$, don't i get a primitive for $f$?

thanks

  • When you did the derivative to get that $F'(z)=f(z)$ you have already assumed that $F(z)$ is the series expansion about $a$. Defining $F$ differently (as you want to do) near $a$ might lead to problems as $F$ might become non-differentiable. – Hasan Saad May 01 '16 at 14:03
  • @HasanSaad so the problem is that for example two "nearby" series expansions may be different so my $F$ is no more well defined? –  May 01 '16 at 14:09
  • @HasanSaad and is the claim true at all? –  May 01 '16 at 14:09
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    That's how I imagined it, the issue I mean. The claim is true for simply connected sets. – Hasan Saad May 01 '16 at 14:10
  • you proved that a primitive exists on a disk where $f$ is analytic, hence by analytic continuation on any path of such disks. but when the path makes a loop, there is a problem, you might not find the same function as you had previously (branch points). that's why you have to consider simply connected open sets, where the Cauchy integral theorem ensures that you'll get the same function (since $\int_C f(z) dz = 0$ for any $C$ in the simply connected open) – reuns May 01 '16 at 14:46

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Since the holomorphic function $f(z) = 1/z$ fails to have a holomorphic primitive on the punctured plane $D = \mathbf{C} \setminus\{0\}$, there must be a flaw somewhere in the proposed argument. That somewhere is at the end, the implicit assertion that the power series (plural) defining $F$ in a neighborhood of an arbitrary point "patch together" globally.

For further reading, the umbrella for this type of consideration is analytic continuation. If you're algebraically-minded, sheaf cohomology is a powerful, sophisticated bookkeeping device "designed" almost expressly for this type of investigation.

Andrew D. Hwang
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