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 $z>1$.
 A: The Mathematica 11.3 command
Series[Hypergeometric2F1[a, b, a + b, z], {z, 1, 1}, Assumptions -> z > 1]

produces
$$-\frac{\Gamma (a+b) (\psi ^{(0)}(a)+\psi ^{(0)}(b)+\log (z-1)+i \pi +2 \gamma )}{\Gamma (a) \Gamma (b)}+\frac{(z-1) \Gamma (a+b) (a b \log (z-1)-2 a b+i \pi  a b+2 \gamma  a b+a b \psi ^{(0)}(a+1)+a b \psi ^{(0)}(b+1))}{\Gamma (a) \Gamma (b)}+O\left((z-1)^2\right) $$
The Maple 2018 command
series(hypergeom([a, b], [a+b], z), z = 1, 2) assuming z>1

performs
$$-{\frac {\Gamma \left( a+b \right)  \left( \ln  \left( z-1 \right) +i
\pi+2\,\gamma+\Psi \left( a \right) +\Psi \left( b \right)  \right) }{
\Gamma \left( a \right) \Gamma \left( b \right) }}+{\frac {\Gamma
 \left( a+b \right) ab \left( \ln  \left( z-1 \right) +i\pi-2+2\,
\gamma+\Psi \left( a+1 \right) +\Psi \left( b+1 \right)  \right) 
 \left( z-1 \right) }{\Gamma \left( a \right) \Gamma \left( b \right) 
}}+O \left(  \left( z-1 \right) ^{2} \right) .
 $$ 
Numerical calculations (on demand) for $a=1,b=1/2,z=1.01$ confirm both the Mathematica's result and the Maple's result.
A: Mathematica says:
$$\left(\frac{\pi  \csc ((-a-b+c) \pi ) \Gamma (c)}{\Gamma (a+b-c+1) \Gamma (c-a) \Gamma
   (c-b)}-\frac{a b \pi  \csc ((-a-b+c) \pi ) \Gamma (c) (z-1)}{\Gamma (a+b-c+2) \Gamma
   (c-a) \Gamma (c-b)}+\frac{a (a+1) b (b+1) \pi  \csc ((-a-b+c) \pi ) \Gamma (c) (z-1)^2}{2
   \Gamma (a+b-c+3) \Gamma (c-a) \Gamma (c-b)}+O\left((z-1)^3\right)\right)+(z-1)^{-a-b+c}
   e^{2 i \pi  (-a-b+c) \left\lfloor -\frac{\arg (z-1)}{2 \pi }\right\rfloor }
   \left(\frac{(-1)^{-a-b+c+1} \pi  \csc ((-a-b+c) \pi ) \Gamma (c)}{\Gamma (a) \Gamma (b)
   \Gamma (-a-b+c+1)}+\frac{(-1)^{-a-b+c+1} (a-c) (c-b) \pi  \csc ((-a-b+c) \pi ) \Gamma (c)
   (z-1)}{\Gamma (a) \Gamma (b) \Gamma (-a-b+c+2)}-\frac{\left((-1)^{-a-b+c} (a-c-1) (a-c)
   (c-b) (-b+c+1) \pi  \csc ((-a-b+c) \pi ) \Gamma (c)\right) (z-1)^2}{2 (\Gamma (a) \Gamma
   (b) \Gamma (-a-b+c+3))}+O\left((z-1)^3\right)\right)$$
