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I am interested in obtaining the asymptotic expansion of $r(\rho)$ (which is the inverse of the object rho[r_,b_,q_] below). Basically I want to series expand rho[r_,b_,q_] for large $r$ (i.e. as $r\to \infty$) and then invert the series to obtain $r(\rho)$. I have tried the $\textbf{InverseSeries}$ function in Mathematica but I am not really sure if I am doing the right thing. By the way, $b$ is just some positive constant while $-\infty<q<1$.

rho[r_,b_,q_]:=(2b/(1-q))(1-(b/r)^(1-q))^(1/2)Hypergeometric2F1[1/2,1-1/(q-1),3/2,1-(b/r)^(1-q)]
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