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Bug introduced in 9.0 and fixed in 10.0.0

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, 

Mathematica 9:

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

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

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With eqn = Integrate[x^(1/2)/( Exp[x - \[Mu]] - 1), {x, 0, Infinity}] it looks like eqn is always positive so the equation has no solution. – b.gatessucks Jan 26 '13 at 16:57
@b.gatessucks: True. Original there was a bunch of user constants. Just a mistyped after == sign. – m0nhawk Jan 26 '13 at 20:09
This seems to be fixed in V10 (although the output is now in terms of Root instead of InverseFunction). – Michael E2 Aug 21 '15 at 17:40
up vote 9 down vote accepted

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]]

enter image description here

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