Is the union of primitive open sets not primitive, unless their intersection is connected? We say an open set $\Omega$ is primitive if every holomorphic function in $\Omega$ has a primitive in $\Omega$. I have 2 questions about primitive sets.

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*I am looking for a counterexample that shows that the union of two primitive sets is not always primitive

*I am trying to prove that, if the intersection of two primitive sets is connected, then their union is primitive

For 1. I used two open disks with radius $a$, one with centre $-a$ and the other with centre $a$. I know in the two disks $f(z)=\frac{1}{z}$ has a primitive but I think that in the union of the two disks $f(z)$ does not have a primitive.
Any help would be welcome.
Thank you in advance.
 A: Your counterexample doesn't work, since $\frac{1}{z}$ does have a primitive on the union of the two sets. The two sets are disjoint, so you can just take the primitives on each of the sets and define a primitive on their union by Combining the two. No problems will arise, since their domains don't overlap.
To find a counterexample, you should first be clear about which sets are primitive, and which ones are not. For instance, star domains are primitive according to Cauchy's integral theorem. But pointed sets like $D\backslash\{a\}$ where $D$ is a domain and $a\in D$ are not primitive, since $\frac{1}{z-a}$ has no primitive. Considering this, if you find two star domains whose union is a pointed domain, you got your counterexample. For instance, take $\mathbb C\backslash[0,\infty)$ and $\mathbb C\backslash(-\infty,0]$, whose union is $\mathbb C\backslash\{0\}$.
To your second question: Let $A,B$ be primitive, $A\cap B$ connected, and $f$ holomorphic on $A\cup B$. Then $f$ has a primitive $F_A$ on $A$, and a primitive $F_B$ on $B$. Both of these are primitives on $A\cap B$, and since that's a connected set, they only differ by a constant on it. Now adjust one of the primitives by adding or subtracting that constant to make them both equal on $A\cap B$. Now just glue $F_A$ and $F_B$ together to obtain a primitive on $A\cup B$. No problems will arise, this time because they are equal on the overlap of their domains.
