
Solve[x^2(x^2-1)==e,x]
bb = x /. Last[%]
aa=Sqrt[e]/bb
alpha=Sqrt[2628.8256429]

Integrate[Sqrt[(a^2+x^2)(b^2-x^2)],{x,-b,b}]
Simplify[%,{a>0,b>0}]

Out[6]=(2*a*((-a^2+b^2)*EllipticE[-(b^2/a^2)]+(a^2+b^2)*EllipticK[-(b^2/a^2)]))/3

% /. {a->aa,b->bb}
Simplify[%]
wkb=alpha * %
Plot[wkb/Pi,{e,0,.3}]
Export["346t304#1WKB.pdf",%]

FindRoot[wkb/Pi==21.5,{e,.2}]
results=Table[FindRoot[wkb==(n-.5)*Pi,{e,.1}],{n,12,22}]

{{e -> 0.00896469}, {e -> 0.0270078}, {e -> 0.0474129}, 
{e -> 0.0695849}, {e -> 0.0932427}, {e -> 0.118214}, {e -> 0.144377}, 
{e -> 0.171643}, {e -> 0.199938}, {e -> 0.229204}, {e -> 0.25939}}

Integrate[Sqrt[(x^2-a^2)(b^2-x^2)],{x,a,b}] !NoJoy
(b*((a^2+b^2)*EllipticE[(b^2-a^2)/b^2]-2*a^2*EllipticK[(b^2-a^2)/b^2]))/3
% /. {a->Sqrt[-e]/bb,b->bb}
Simplify[%]
wkb2=alpha * %
Plot[wkb2/Pi,{e,-.25,0}]
Export["346t304#1WKB2.pdf",%]

FindRoot[wkb2/Pi==.5,{e,-.2}]
results=Table[ FindRoot[Re[wkb2]==(n-.5)*Pi,{e,-.1}],{n,1,5}]
 {{e -> -0.222709}, {e -> -0.169991}, {e -> -0.120095}, 
 {e -> -0.073624}, {e -> -0.0317139}}


