I am interested in the large $x$ asymptotics for the function $$ \mathrm{Re}\left\{ \ _2F_1\left(\tfrac{1}{2}, \tfrac{5}{2} ; 2 ; x \right) \ \right\} $$ When I check the series expansion at $x = \infty$ Mathematica tells me that I should expect: $$ \ _2F_1\left(\tfrac{1}{2}, \tfrac{5}{2} ; 2 ; x \right) = \left[ -\frac{1}{4} x^{-5/2} \right] + i \left[ - \frac{8}{3\pi} x^{-1/2} - \frac{2}{3\pi} x^{-3/2} + \frac{\left[ 6 \log(8x) - 19 \right]}{24\pi} x^{-5/2} \right] + \mathscr{O}(x^{-7/2}) $$ which I think follows from the asymptotic series given in this link (the first entry in the first formula where $b-a = 2 \in \mathbb{N}^{+}$ and $c - a = \tfrac{3}{2} \notin \mathbb{Z}$ for this function). This means that taking the real part of the above, I should get for large $x$ $$ \mathrm{Re} \{ \ _2F_1\left(\tfrac{1}{2}, \tfrac{5}{2} ; 2 ; x \right) \ \} \approx -\frac{1}{4} x^{-5/2} + \mathscr{O}(x^{-7/2}) $$
My Issue: These asymptotics do not match in Mathematica by a huge amount. For example, picking $x = 10^{100}$ I find that:
It seems that there is a disagreement here!
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What is going on? I wonder if it has to do with the fact that $_2F_1$ has a branch cut along $x \in (1,\infty)$? I find that even if I add a tiny complex piece to the argument (to dodge the branch cut), that I am not getting agreement.