NSolve: Mathematica 9 issue

Question

Bug introduced in 9.0 and fixed in 10.0.0

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In Mathematica 8 the following expression evaluates and returns a result in terms of InverseFunction:

NSolve[Integrate[x^(1/2)/(Exp[x - μ] - 1), {x, 0, ∞}] == C, μ]

where C is a constant. But in Mathematica 9 the expression does not evaluate:

NSolve::nsmet: This system cannot be solved with the methods available to NSolve. >>

Is this a bug, a regression, or a feature? And how can this equation can be solved in V9?

Particular, trying to solve this:

NSolve[PolyLog[3/2, 20 x] == 2, x]

In Mathematica 8 and Mathematica 9 gives different results:

Mathematica 8 (correct result):

NSolve::ifun: Inverse functions are being used by NSolve, so some solutions may not be found; use Reduce for complete solution information. >>
{{x -> 0.05 InverseFunction[PolyLog, 2, 2][3/2, 
 2.0000000000000000000000000000000]}}

Mathematica 9:

NSolve::nsmet: This system cannot be solved with the methods available to NSolve. >>
NSolve[PolyLog[3/2, 20 x] == 2, x]

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UPD: That's a confirmed bug with NSolve, that I already reported to developers of Wolfram Mathematica long time ago.

Answer 1

In general for a non-polynomial equation you can use FindRoot to solve it :

eqn[m_] = Integrate[x^(1/2)/( Exp[x - m] - 1), {x, 0, \[Infinity]}] ;

(* using 2 initial values for the solver *)
roots[r_?NumericQ] := FindRoot[eqn[m] == r, {m, -1.23, -0.456}][[1, 2]]

Check :

dataRoots = {roots[#], #} & /@ Range[0.1, 2, 0.1];

Show[ListPlot[dataRoots, PlotStyle -> {Red, PointSize[0.02]}, 
           GridLines -> Transpose[dataRoots], AxesOrigin -> {0, 0}], 
     Plot[eqn[m], {m, -10, 0}, PlotRange -> All]]