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I am to solve for $r(\rho)$ given the function,

ρAsymp[r_, b_, q_] := 
  1/(1 - q) Gamma[1/(1 - q)]/Gamma[(q - 2)/(q - 1)] r Sqrt[1 - (b/r)^(1 - q)]

This can be solved in a straightforward way by converting this into a quadratic equation and use Solve to find for the root corresponding to $r(\rho)$ for every value of $q$. But I have difficulty implementing the InverseFunction to this problem.

Can someone help me with this? Also, I want to scan $r(\rho)$ for all values of $q<1$, how can I implement this by using a Module?

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  • $\begingroup$ Recommend that you change your definition to \[Rho]Asymp[r_, b_, q_] := 1/(1 - q) Gamma[1/(1 - q)]/Gamma[(q - 2)/(q - 1)] r Sqrt[ 1 - (b/r)^(1 - q)] // FullSimplify // Evaluate which will be simplified to Sqrt[1 - (b/r)^(1 - q)]*r $\endgroup$
    – Bob Hanlon
    Commented Jan 15, 2019 at 15:55
  • $\begingroup$ I did not see that. Thank you for pointing out. Is there an efficient way of getting $r(\rho)$? for all allowed values of $q$? $\endgroup$
    – user583893
    Commented Jan 16, 2019 at 2:35
  • $\begingroup$ Have you tried using FullSimplify on your function? The gamma functions all disappear and I get ρ = r*Sqrt[1-(b/r)^(1-q)]. Doesn't solve your problem though. $\endgroup$
    – Roman
    Commented Jun 14, 2019 at 9:13

2 Answers 2

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Looking at the examples in the documentaion you could do the following:

g = InverseFunction[Function[{r, b, q}, ρAsymp[r, b, q]], 1, 3]

which gives the inverse of ρAsymp[r, b, q], a function of 3 arguments, with respect to it's first argument r. Evaluation is then

g[1, 2, 3]

Sqrt[2]

The function g may now be used as a usual function.

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  • $\begingroup$ What about symbolically? If i only specify, $b$ and $q$ $\endgroup$
    – user583893
    Commented Jan 15, 2019 at 7:43
  • $\begingroup$ @user583893 sadly that doesn't work for this particular function and my suggested solution. $\endgroup$
    – gothicVI
    Commented Jan 15, 2019 at 8:12
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This function shows a very different behavior depending on parameters. Decide which parameter combinations are relevant for you with the help of a graphical overview.

Manipulate[
  ParametricPlot[{Sqrt[1 - (b/r)^(1 - q)] r /. q -> s/t, r}, {r, -10, 10}, 
    AxesLabel -> {"rho", "r"}, AspectRatio -> 1], 
  {{b, -2}, -5, 5}, 
  {{s, 3}, Join[Range[-5, 5], {1/Pi, Pi, 10 Pi, -1/Pi, -Pi, -10 Pi}]}, 
  {t, Range[5]}]

enter image description here

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