We all know that the famous Hypergeoemtric function $\_2F\_1$ has an integral form as follows:


$$\_2F\_1(a,b,c;z)=-\frac{e^{-i\pi c} \Gamma(c)}{4\Gamma(b)\Gamma(c-b)\sin\pi b\sin\pi(c-b)}\int\_P t^{b-1}(1-t)^{c-b-}(1-zt)^{-a}dt,$$


where $c\neq0,-1,-2,...$ and $P$ is Pochhammer contour. This integral form can give an analytic continuation of Hypergeometric function from $|z|<1$ to the whole $z$-plain except the slit $[1,\infty]$.


Recently, I read a paper -


* W. Becken, P. Schmelcher, *The analytic continuation of the Gaussian hypergeometric function $\_2F\_1(a,b,c;z)$ for arbitrary parameters*, Journal of Computational and Applied Mathematics **126** (2000) 449–478. <https://doi.org/10.1016/S0377-0427(00)00267-3>, <https://core.ac.uk/download/pdf/82108003.pdf>


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, and I can regard the hypergeometric function as an entire function except two exact points. Can I directly use this conclusion? Is this conclusion verified formally in maths world? Is the continuated function single-valued?



Does any Hyper-geometric function can be analytically continuated to the whole complex plain except $e^{\pm i\pi/3}$
Continuation to the whole $z$-plane? As stated, not true. Example:
$$
{}\_2F\_1\left(-\frac12,1;1;z\right) = (1-z)^{1/2},\quad |z|<1,
$$
cannot be extended analytically to any neighborhood of $z=1$.


