Is there a rigorous way of inverting


where $B(1-(\frac{b_{0}}{r})^{1-q},\frac{1}{2},\frac{1}{q-1})$ is an incomplete beta function, $b_{0}$ is some positive constant, $-\infty<q<1$. Basically I want to obtain $r(\rho)$. I am thinking of getting the asymptotic expansion of $\rho(r)$ (since I am interested in the regime where $r$ is very large) and then invert using Lagrange inversion method. Maybe I only need the first three terms of the expansion. Am I allowed to do this? Or is there a more convenient way of inverting the given function containing an incomplete beta function?


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