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Re: FriCAS failure on Goursat(?) pseudoelliptics
Posted:
Sep 29, 2012 4:21 PM


Waldek Hebisch schrieb: > > clicliclic@freenet.de wrote: > > > > clicliclic@freenet.de schrieb: > > > > > > Waldek Hebisch schrieb: > > > > > > > > clicliclic@freenet.de wrote: > > > > > > > > > > One of the pseudoelliptic examples (here taken from Goursat's > > > > > Cours d'analyse) is given on page 50 of the booklet mentioned. > > > > > > > > AFAICS this example is wrong  FriCAS result indicates that > > > > there is no elementary integral in this case and I see no > > > > reasons why it should have one. OTOH, if one modifies > > > > the example to have > > > > > > > > f/sqrt((1  x^2)*(1  k^4*x^2)) > > > > > > > > under integral, then FriCAS can handle it for particular fs. > > > > > > > > > > Thanks for trying. My GradshteynRyzhik has three such integrands and > > > adds more detail (without specifying the source). In particular, the > > > substitution z*x = SQRT(x*(1x)*(1k^2*x)) is said to turn the > > > integrand into a rational function. I will look into this claim. The > > > other two examples are the same integrand f(x)/sqrt(x*(1x)*(1k^2*x)) > > > with f(x) + f((1k^2*x)/(k^2*(1x))) = 0 and with > > > f(x) + f((1x)/(1k^2*x)) = 0. > > > > > > > If you made no mistake, FriCAS is in trouble here: I have confirmed that > > the first example (the one cited by Hardy) is indeed elementary, using > > the rational factor f(x) = (k^2*x^2  1)/((a*k^2*x + b)*(b*x + a)) for > > which f(x) + f(1/(k^2*x)) = 0 as required. > > I made a mistake: I took the irrational factor to be > sqrt(x*(1  x)*(1  k^2*x^2)). Goursat example is clearly > based on symmetry under x > 1/(k^2*x) map, but that factor > is not symmetric under such map, so has no reason to have > elementary integral. When irrational factor is changed to > sqrt((1x^2)*(1  k^2*x^2))) we get required symmetry, and > it works fine. I did not notice that replacing k^2*x^2 by > k^2*x also gives symmetry (and that this is what text > actually contained). > > To make things clear: with sqrt(x*(1  x)*(1  k^2*x)) FriCAS > can do the integrals. >
Excellent. So a computation of the "splitting field of the Trager resultant" is not needed here. Goursat was clearly an authority in pseudoelliptic integrals, as shown by this 1887 paper of his:
<http://www.numdam.org/item?id=BSMF_1887__15__106_1>
Martin.



