Primitive function of $ \frac{1}{z} $ Let $ U_1,U_2,U_3,U_4 \subset \mathbb{C} $ be the upper, lower,right and left open half-planes with respect to the axes (for example $ U_{1}=\left\{ z:Im\left(Z\right)>0\right\}  $ ).
Show that the function $ f(z) =  \frac{1}{z} $ has a primitive on each union of one, two or three of the half-planes $U_i $, but not on the union of all four.
Im not sure how to find $ F $ such that for all, say $ z\in U_{1}\cup U_{2}\cup U_{3} $, the following holds:$$ F'\left(z\right)=\frac{1}{z} $$
What I do know how to do, is for each, say, $ a\in U_{1}\cup U_{2}\cup U_{3} $, to find a neigborhood where $f(z) $ has primitive, but that is not what I need.
I would highly appreciate help here.
Thanks in advance.
 A: You just need to show there exists a branch of logarithm in these domains. For example, let's look at $U_1\cup U_2\cup U_3$, which is just $\mathbb{C}\setminus (-\infty, 0]$. In this domain we have the principal branch of the argument function, which is the argument in the interval $(-\pi, \pi)$. This principal branch is usually denoted by $Arg(z)$. This allows us to define a branch of logarithm $Log: \mathbb{C}\setminus (-\infty, 0]\to\mathbb{C}$ by:
$Log(z)=ln|z|+iArg(z)$
It is well known that branches of logarithm are holomorphic, and the derivative is $\frac{1}{z}$.
A: Here is a general argument.
Primitives (of a holomorphic function $f : U \to \mathbb C$ on a connected subdomain of $U$) are unique up to an additive constant. So if you know primitives $F_1$, $F_2$, $F_3$ on connected domains $U_1$, $U_2$, $U_3 \subset U$, and if $U_1 \cup U_2 \cup U_3$ is connected, and if you know that a primitive exists on $U_1 \cup U_2 \cup U_3$ (for example if you know that $U_1 \cup U_2 \cup U_3$ is simply connected), then you can compute a formula for a primitive on $U_1 \cup U_2 \cup U_3$. The formula will have the form
$$F(z) = 
\begin{cases}
F_1(z) + C_1 &\quad\text{if $z \in U_1$} \\
F_2(z) + C_2 &\quad\text{if $z \in U_2$} \\
F_3(z) + C_3 &\quad\text{if $z \in U_3$}
\end{cases}
$$
but the constants $C_1,C_2,C_3$ must be specified. You can choose $C_1$ arbitrarily (e.g. $C_1=0$), and then $C_2$ and $C_3$ are determined. You can compute $C_2,C_3$ by exploiting the fact that at least two of the three intersections $U_1 \cap U_2$ or $U_2 \cap U_3$ or $U_3 \cap U_1$ are nonempty. For instance, if there exists $w \in U_1 \cap U_2$ then $F(w)=F_1(w)+C_1=F_2(w)+C_2$ and therefore $C_2 = F_1(w)-F_2(w)+C_1$.
