Pensieve header: Computing ν to degree 6 following http://katlas.math.toronto.edu/drorbn/bbs/show?shot=Dancso-120430-110839.jpg.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2012-05\\beta5.0"];
<< betaCalculus.m

Clear[ℏ];
$PerturbativeDegree=6;
βSimplify[expr_]:= Replace[
Series[Normal[expr], {ℏ, 0, $PerturbativeDegree}],sd_SeriesData :> MapAt[Expand, sd, 3]
];
βCollect[B[ω_, μ_]] := B[
βSimplify[ω],
βSimplify[μ]
];

{V1, C1, sol} = Get["SolutionToDegree6-120501.m"];

Φ1=Φ[V1]

{
Φ1 // dS[2] // dm[3,2,2] // dm[2,1,1] ,
Φ1 // dS[2] // dm[3,2,2] // dm[2,1,1] // Inverse
}

(1+(1/24) (Subscript[c, 1]^2) (ℏ^2)+(((Subscript[c, 1]^4) (ℏ^4))/1920)+(((Subscript[c, 1]^6) (ℏ^6))/322560)+((O[ℏ])^7))(1-(1/24) (Subscript[c, 1]^2) (ℏ^2)+((7 (Subscript[c, 1]^4) (ℏ^4))/5760)-((31 (Subscript[c, 1]^6) (ℏ^6))/967680)+((O[ℏ])^7))

Series[(Sqrt[((Sinh[x/2])/(x/2))]), {x,0,6}]

Series[((Sinh[x/2])/(x/2)), {x,0,6}]