We all know that the famous Hypergeoemtric function has an expression form of ordinary differential equation as follows:
$$z(1-z)\frac {d^2w}{dz^2} + \left[c-(a+b+1)z \right] \frac {dw}{dz} - ab\,w = 0.$$
In the ODE above we can see that the hypergeometric function has three singular points, $0$ ,$1$ and $\infty$, which means I can analytically continuate the hypergeometric function to the domain of $\{|z|<1\}\cup\{0<|z-1|<1\}\cup\{|z|>1\}\cup\{0<|1-1/z|<1\}$.
Recently, I read a paper - The analytic continuation of the Gaussian hypergeometric function $_2F_1(a,b,c;z)$ for arbitrary parameters. The paper gave us a conclusion that Gaussian hypergeometric function can be analytically continuated to the whole complex $z$-plain excluding only $e^{\pm i\pi/3}$. If the conclusion is true, it will help me a lot since the continuation domain covers nearly a whole plain. Can I directly use this conclusion? Is this conclusion verified formally in maths world? Is the continuated function single-valued?
Thanks for your advice.
p.s. The weblink of the paper I mentioned above is as follow: