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Re: Goursat pseudo-elliptics and the Wolfram Integrator
Posted:
Dec 15, 2012 3:43 PM
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"Nasser M. Abbasi" schrieb:
>
> On 12/14/2012 3:47 PM, clicliclic@freenet.de wrote:
> >
> > I was curious how Mathematica would handle the pseudo-elliptic integrals
> > recently referred to on this newsgroup. So I plugged two examples into
> > the Wolfram Integrator (which presumably represents Mathematica 8):
> >
> > <http://integrals.wolfram.com/index.jsp>
> >
> > Example 1 (cubic radicand):
> >
> > Integrate[(k*x^2 - 1)/((a*k*x + b)*(b*x + a)
> > *Sqrt[x*(1 - x)*(1 - k*x)]), x]
> >
> > ... eeeek! The Integrator replies in terms of incomplete elliptic F,
> > incomplete elliptic Pi, and the imaginary unit. But the antiderivative
> > just is:
> >
> > 2/(Sqrt[a*b]*Sqrt[(a + b)*(a*k + b)])
> > *ArcTan[Sqrt[a*b]*Sqrt[x*(1 - x)*(1 - k*x)]
> > /(Sqrt[(a + b)*(a*k + b)]*x)]
> >
>
> I tried this on version 9 of Mathematica. I get the same as with
> the above web site you mention.
>
> But when I plot the result for some values of a,b,k and compare with
> what you have above, the plots do not look the same? Not even a scaled
> shift, but they look different (near origin). Away from origin, they
> seem to become closer to each others. (i.e. for very large x)
> Here is the results to see:
>
> http://12000.org/my_notes/misc_items/121512/integrals.html
To my eyes, the plotted results seem to agree up to a piecewise constant
- so the Mathematica result should be ok since differentiation on Derive
confirms the ArcTan antiderivative without problem.
>
> I tried it on Maple16. Was not able to plot the result in
> Maple, I get an empty plot. But Maple also gives result in
> terms of incomplete elliptic functions also.
>
> > Example 2 (quartic radicand):
> >
> > Integrate[(k*x^2 - 1)/((a*k*x + b)*(b*x + a)
> > *Sqrt[(1 - x^2)*(1 - k^2*x^2)]), x]
> >
> > ... "Mathematica could not find a formula for your integral. Most likely
> > this means that no formula exists." Waouw! Here the elementary
> > antiderivative is:
> >
> > 2/(Sqrt[(a + b)*(a*k + b)]*Sqrt[(a - b)*(a*k - b)])
> > *ArcTanh[Sqrt[(a + b)*(a*k + b)]*Sqrt[(1 - x^2)*(1 - k^2*x^2)]
> > /(Sqrt[(a - b)*(a*k - b)]*(1 - x)*(1 - k*x))]
> >
>
> I tried this also, yes, Mathematica 9 does not do it.
>
> Maple16 does solve it. It gives answers in terms of incomplete
> elliptic as well. result is in the above link as well.
So Maple wins! The treatment of elliptic integrands in Mathematica is
really ridiculous.
>
> > The theory behind these integrals is given in: Edouard Goursat, Note sur
> > quelques intégrales pseudo-elliptiques, Bulletin de la Société
> > Mathématique de France 15 (1887), 106-120, on-line at:
> >
> > <http://www.numdam.org/item?id=BSMF_1887__15__106_1>
> >
> > This was written 125 years ago - apparently too recent for the "Risch"
> > integrator of Mathematica 8. I expect that FriCAS can do the second
> > integral too. How do Maple and Sympy behave?
> >
>
> Do not use sympy, may be someone who has that can try them.
>
Thanks for the results! And I hadn't known MMA9 was out.
Martin.
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