Consider $f(z)=\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}z^{2n+1}$.
(i) Determine the dominion of $f$.
(ii) Let $\Gamma=\{iy, y\in\mathbb R, |y|\geq 1\}$, and prove that $f$ can be prologued analytically to $\mathbb C\setminus \Gamma$.
(iii) Discuss whether there exist other maximal analytic extensions to domains of the Riemann sphere of the form $(\mathbb > C\cup\{\infty\})\setminus \Gamma'$, with $\Gamma'$ compact in $\mathbb > C$.
The dominion of $f$ should be the open unit disk in $\mathbb C$, in which $f$ converges absolutely, and some points on the unit circle like $1$, for Leibniz (but not $-1$: how can I check wether there are any others? Is there some restatement of Leibniz criterion for complex series?)
For the extension, I tried to use some form of the Schwarz simmetry principle, but, I didn't obtain anything. One could extend to $\mathbb C\setminus S^1$ by putting $f(z)=f(\frac{1}{\bar z})$ for $|z|>1$?
Thank you in advance for answers and suggestion of useful tools.
Recall that
$$\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}x^{2n+1} = \arctan x$$
for $x\in (-1,1).$ So you have to figure that $\arctan z$ is going to be involved here. And so it is: Verify that
$$\tan z : \{-\pi/2 < \text { Re }z < \pi/2\} \to \Gamma$$
is a holomorphic bijection. (This is a nice exercise if you haven't done it before.) It follows that the inverse of $\tan z,$ which we'll call $\arctan z,$ is a biholomorphic map of $\Gamma$ onto the strip above. This is the desired analytic extension of $\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}z^{2n+1}$ to $\Gamma.$