       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)]*(Sqrt[3*(7 -
4*I*Sqrt)]*EllipticPi[1 + I/Sqrt,
ArcSin[Sqrt[(1/3)*I*(2*I + Sqrt)]], (1/3)*(1 + 2*I*Sqrt)]
+
3*I*(I + Sqrt)*(EllipticF[ArcSin[Sqrt[-1 - I/Sqrt]], 1/3 -
(2*I*Sqrt)/3] -
EllipticPi[2/3 - (I*Sqrt)/3, ArcSin[Sqrt[-1 - I/Sqrt]], 1/3
- (2*I*Sqrt)/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:=
> 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//InputForm=
> (8*Sqrt[4 - I*Sqrt]*EllipticF[ArcSin[Sqrt[5*I + Sqrt]/2^(3/4)],
> (2*Sqrt)/(2*I + Sqrt)] -
>    (7*I)*Sqrt[8 - (2*I)*Sqrt]*EllipticF[ArcSin[Sqrt[5*I + Sqrt]/
> 2^(3/4)], (2*Sqrt)/(2*I + Sqrt)] +
>    (12*I)*Sqrt[4 + I*Sqrt]*EllipticF[I*(Log - Log[Sqrt[2 -
> (2*I)*Sqrt] + I*2^(1/4)*Sqrt[2*I + Sqrt]]),
>      (2*Sqrt)/(2*I + Sqrt)] - 3*Sqrt[8 + (2*I)*Sqrt]*
>     EllipticF[I*(Log - Log[Sqrt[2 - (2*I)*Sqrt] +
> I*2^(1/4)*Sqrt[2*I + Sqrt]]), (2*Sqrt)/(2*I + Sqrt)] +
>    9*Sqrt[2*(-8 + (7*I)*Sqrt)]*EllipticPi[(2*Sqrt)/(-I +
> Sqrt), ArcSin[Sqrt[5*I + Sqrt]/2^(3/4)],
>      (2*Sqrt)/(2*I + Sqrt)] - 9*Sqrt[2*(-8 +
> (7*I)*Sqrt)]*EllipticPi[(2*Sqrt)/(-I + Sqrt),
>      I*(Log - Log[Sqrt[2 - (2*I)*Sqrt] + I*2^(1/4)*Sqrt[2*I +
> Sqrt]]), (2*Sqrt)/(2*I + Sqrt)])/
>   (3*Sqrt[-1 + I/Sqrt]*Sqrt[-8 + (7*I)*Sqrt]*(-I + Sqrt)) +
>  EllipticPi[1 - I/Sqrt, ArcSin[Root[2 + 4*#1^2 + 3*#1^4 & , 3, 0]],
> (1 - (2*I)*Sqrt)/3]*Root[8 + 8*#1^2 + 3*#1^4 & , 4, 0]
>
> In:=
> Chop[N[%,30]]
>
> Out=
> 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:=
> > 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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