<s>[INST] Asymptotic expansion of hypergeometric function near $z=1$ Given the hypergeometric function $_2F_1[a,b,c,z]$
in the interval $z\in(1,\infty)$. What is the proper asymptotic expansion of the aforesaid function near $z=1$, when one is approaching from $z>1$.
 [/INST] Mathematica says:
$$\left(\frac{\pi  \csc ((-a-b+c) \pi ) \Gamma (c)}{\Gamma (a+b-c+1) \Gamma (c-a) \Gamma
   (c-b)}-\frac{a b \pi  \csc ((-a-b+c) \pi ) \Gamma (c) (z-1)}{\Gamma (a+b-c+2) \Gamma
   (c-a) \Gamma (c-b)}+\frac{a (a+1) b (b+1) \pi  \csc ((-a-b+c) \pi ) \Gamma (c) (z-1)^2}{2
   \Gamma (a+b-c+3) \Gamma (c-a) \Gamma (c-b)}+O\left((z-1)^3\right)\right)+(z-1)^{-a-b+c}
   e^{2 i \pi  (-a-b+c) \left\lfloor -\frac{\arg (z-1)}{2 \pi }\right\rfloor }
   \left(\frac{(-1)^{-a-b+c+1} \pi  \csc ((-a-b+c) \pi ) \Gamma (c)}{\Gamma (a) \Gamma (b)
   \Gamma (-a-b+c+1)}+\frac{(-1)^{-a-b+c+1} (a-c) (c-b) \pi  \csc ((-a-b+c) \pi ) \Gamma (c)
   (z-1)}{\Gamma (a) \Gamma (b) \Gamma (-a-b+c+2)}-\frac{\left((-1)^{-a-b+c} (a-c-1) (a-c)
   (c-b) (-b+c+1) \pi  \csc ((-a-b+c) \pi ) \Gamma (c)\right) (z-1)^2}{2 (\Gamma (a) \Gamma
   (b) \Gamma (-a-b+c+3))}+O\left((z-1)^3\right)\right)$$
</s>