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MathGroup Archive 2007

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Re: elliptic integral (reloaded!) (version 4 resullt!)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg75964] Re: elliptic integral (reloaded!) (version 4 resullt!)
  • From: dimitris <dimmechan at yahoo.com>
  • Date: Mon, 14 May 2007 03:37:20 -0400 (EDT)
  • References: <f23ome$n1h$1@smc.vnet.net><f26oo3$531$1@smc.vnet.net>

For anyone interested in, here is the solution by mathematica 4

FullSimplify[Integrate[Sqrt[(1 - x)/((x - 2)*(x^2 - 2*x + 3))], {x, 1,
2}]]

(1/9)*Sqrt[(2/3)*(-2 - 5*I*Sqrt[2])]*(Sqrt[3*(7 -
4*I*Sqrt[2])]*EllipticPi[1 + I/Sqrt[2],
     ArcSin[Sqrt[(1/3)*I*(2*I + Sqrt[2])]], (1/3)*(1 + 2*I*Sqrt[2])]
+
   3*I*(I + Sqrt[2])*(EllipticF[ArcSin[Sqrt[-1 - I/Sqrt[2]]], 1/3 -
(2*I*Sqrt[2])/3] -
     EllipticPi[2/3 - (I*Sqrt[2])/3, ArcSin[Sqrt[-1 - I/Sqrt[2]]], 1/3
- (2*I*Sqrt[2])/3]))

N[%, 30]

0=2E9752615369238655188048453717492992972879577`30. + 0``64.2344*I

NIntegrate[Sqrt[(1 - x)/((x - 2)*(x^2 - 2*x + 3))], {x, 1, 2},
WorkingPrecision -> 60, PrecisionGoal -> 30, MaxRecursion -> 12]

0=2E975261536923865518804845371747769411552590737095`30.189694608876156
+ 0``30.2005735123453*I

I have tried the Newton-Leibniz formula but Mathematica 4 can't get
the limit at the endpoints.

So the question remains why Mathematica 6 and Mathematica 5.2 return
the definite integral
unevaluated whereas our old (but good!) friend Mathematica 4 returns a
closed form result?

Any ideas?

Dimitris

=CF/=C7 dimitris =DD=E3=F1=E1=F8=E5:
> If I was not clear let be a little more specific!
>
> I mean
>
> o//ReleaseHold
>
> produce a result in version 4 which having been checked numerically
> agrees with the relevant numerical estimation (by NIntegrate).
>
> On the other hand
>
> o//ReleaseHold
>
> returns the integral unevaluated in version 5.2.
>
> David Cantrell proposed the following workaround in version 5.2.
>
> In[2635]:=
> Integrate[(x - 1)/Sqrt[(1 - x)*(x - 2)*(x^2 - 2*x + 3)], {x, 1, 3/2}]
> + Integrate[Sqrt[(1 - x)/((x - 2)*(x^2 - 2*x + 3))],
>    {x, 3/2, 2}]//InputForm
>
> Out[2635]//InputForm=
> (8*Sqrt[4 - I*Sqrt[2]]*EllipticF[ArcSin[Sqrt[5*I + Sqrt[2]]/2^(3/4)],
> (2*Sqrt[2])/(2*I + Sqrt[2])] -
>    (7*I)*Sqrt[8 - (2*I)*Sqrt[2]]*EllipticF[ArcSin[Sqrt[5*I + Sqrt[2]]/
> 2^(3/4)], (2*Sqrt[2])/(2*I + Sqrt[2])] +
>    (12*I)*Sqrt[4 + I*Sqrt[2]]*EllipticF[I*(Log[2] - Log[Sqrt[2 -
> (2*I)*Sqrt[2]] + I*2^(1/4)*Sqrt[2*I + Sqrt[2]]]),
>      (2*Sqrt[2])/(2*I + Sqrt[2])] - 3*Sqrt[8 + (2*I)*Sqrt[2]]*
>     EllipticF[I*(Log[2] - Log[Sqrt[2 - (2*I)*Sqrt[2]] +
> I*2^(1/4)*Sqrt[2*I + Sqrt[2]]]), (2*Sqrt[2])/(2*I + Sqrt[2])] +
>    9*Sqrt[2*(-8 + (7*I)*Sqrt[2])]*EllipticPi[(2*Sqrt[2])/(-I +
> Sqrt[2]), ArcSin[Sqrt[5*I + Sqrt[2]]/2^(3/4)],
>      (2*Sqrt[2])/(2*I + Sqrt[2])] - 9*Sqrt[2*(-8 +
> (7*I)*Sqrt[2])]*EllipticPi[(2*Sqrt[2])/(-I + Sqrt[2]),
>      I*(Log[2] - Log[Sqrt[2 - (2*I)*Sqrt[2]] + I*2^(1/4)*Sqrt[2*I +
> Sqrt[2]]]), (2*Sqrt[2])/(2*I + Sqrt[2])])/
>   (3*Sqrt[-1 + I/Sqrt[2]]*Sqrt[-8 + (7*I)*Sqrt[2]]*(-I + Sqrt[2])) +
>  EllipticPi[1 - I/Sqrt[2], ArcSin[Root[2 + 4*#1^2 + 3*#1^4 & , 3, 0]],
> (1 - (2*I)*Sqrt[2])/3]*Root[8 + 8*#1^2 + 3*#1^4 & , 4, 0]
>
> In[2636]:=
> Chop[N[%,30]]
>
> Out[2636]=
> 0=2E975261536923865518804845371749
>
> which it looks quite mysterious to me! (3/2???)
>
> Dimitris
>
>
> =CF/=C7 dimitris =DD=E3=F1=E1=F8=E5:
> > I guess saying many things prevent forumists from answering?!
> >
> > Anyway...
> >
> > In[9]:=
> > o = HoldForm[Integrate[Sqrt[(1 - x)/((x - 2)*(x^2 - 2*x + 3))], {x, 1,
> > 2}]]
> >
> > Version 4. succeeds in getting a closed form result.
> > Version 5.2 returns the integral unevalueated.
> >
> > I just want to know what version 6 does!
> >
> > Thanks a lot!
> >
> > Dimitris



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