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I am trying to solve an equation having degenerate limit of Fermi-Dirac integral.

My code basically is 

ClearAll["Global`*"]
FD1[d_, η_] := η^(d + 1)/
  Gamma[d + 2] ;  (* Defining the Fermi-Dirac integrals in degenerate *)

Solve[FD1[(d-2)/2, ηs] + FD1[(d-2)/2, ηs - vd] == 
  2 FD1[(d-2)/2, η0], {ηs}]

I want to solve for $\eta_S$ in terms of $\eta_0$ and $v_d$. Is it possible to solve it using this method? (And maybe expand the solution in terms of $vd$ as $vd$ is small and we can only pick the first few terms)

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I would recommend using the new in M12 function AsymptoticSolve for this. Your equation:

eqn = FD1[(d-2)/2, ηs] + FD1[(d-2)/2, ηs - vd] == 2 FD1[(d-2)/2, η0];

We need to find the zeroth order approximation of ηs when vd is small:

Simplify[Solve[eqn /. vd -> 0], (η0 | vd) ∈ Reals]

{{ηs -> η0}}

Now, use AsymptoticSolve:

AsymptoticSolve[eqn, {ηs, η0}, {vd, 0, 5}]

{{ηs -> vd/2 + ((-6 + d) (-2 + d) (-1 + d) vd^4)/( 1536 η0^3) + ((2 - d) vd^2)/(16 η0) + η0}}

Addendum

If you have an earlier version of Mathematica, so that you don't have access to AsymptoticSolve, you could try using the cloud instead. For example, define:

asymptoticSolve[args__] := CloudEvaluate[System`AsymptoticSolve[args]]

Then use asymptoticSolve instead of AsymptoticSolve.

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  • $\begingroup$ Thanks a lot, thats really helpful. I don't have M12 so I cannot use AsymptoticSolve. Is there any other similar function in previous versions? $\endgroup$ Commented Sep 12, 2019 at 15:59
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    $\begingroup$ @Indeterminate What does CloudEvaluate[$VersionNumber] return for you? $\endgroup$ Commented Sep 12, 2019 at 16:01
  • $\begingroup$ It returns 12. But somehow, AsymptoticSolve doesn't work. $\endgroup$ Commented Sep 12, 2019 at 16:03
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    $\begingroup$ @Indeterminate See update. $\endgroup$ Commented Sep 12, 2019 at 16:08
  • $\begingroup$ Thanks a lot. It works. $\endgroup$ Commented Sep 12, 2019 at 16:10

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