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Topic: can your system handle 4th-root pseudo-elliptics?
Replies: 17   Last Post: Sep 24, 2017 12:37 PM

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 Albert D. Rich Posts: 311 From: Hawaii Island Registered: 5/30/09
Re: can your system handle 4th-root pseudo-elliptics?
Posted: Sep 23, 2017 4:41 PM

On Saturday, September 23, 2017 at 7:37:49 AM UTC-10, clicl...@freenet.de wrote:
> Albert Rich schrieb:
> >
> > On Friday, September 22, 2017 at 6:58:04 AM UTC-10, clicl...@freenet.de wrote:

> > >
> > > Albert Rich schrieb:

> > > >
> > > > As Martin pointed out, when m=0 and n=1 or when m=2 and n=3 these
> > > > integrals are pseudo-elliptic. However for other even values of
> > > > m, the optimal antiderivatives do seem to require a single, simple
> > > > elliptic term. Do you concur?

> > >
> > > I suppose this is most easily answered by means of a capable
> > > algebraic Risch integrator.

> >
> > For the antiderivative of x^2/((2-x^2)*(x^2-1)^(1/4)) Rubi 4.13.3 gets
> >
> > ArcTan[x/(Sqrt[2]*(x^2-1)^(1/4))]/Sqrt[2] +
> > ArcTanh[x/(Sqrt[2]*(x^2-1)^(1/4))]/Sqrt[2] -
> > 2*(1-x^2)^(1/4)/(x^2-1)^(1/4)*EllipticE[ArcSin[x]/2,2]
> >
> > involving only a single EllipticE function. If this was actually a
> > seudo-elliptic integral, that would imply EllipticE[ArcSin[x]/2,2]
> > could be expressed in terms of elementary functions. But surely that
> > is not the case(?). Ergo the integral is elliptic.
> >

>
> I too believe that E(phi, k) := INT(SQRT(1 - k^2*SIN(p)^2), p, 0, phi)
> cannot be expressed in terms of elementary function unless k^2 = 0 or
> k^2 = 1, but I cannot name a source for this. The Digital Library of
> Mathematical Functions at <http://dlmf.nist.gov/> may be a good point
> to start digging. For specific values of k^2, however, an on-line proof
> can be ordered at:
> <http://axiom-wiki.newsynthesis.org/FriCASIntegration?root=FriCAS>
> Scroll down to the bottom of the page, enter:
>
> \begin{axiom}
> setSimplifyDenomsFlag(true)
> integrate(your_elliptic_integrand, your_integration_variable)
> \end{axiom}
>
> into the grey text box, and hit the Preview Button. After the screen
> has been updated, inspect the result: if your integral is returned
> unevaluated, FriCAS claims that it cannot be expressed in terms of
> elementary functions.
>
> Martin.
>
> PS: Use lower-case sqrt(), sin(), etc.

According to Mathematica, 1/(2*(1-x^2)^(1/4)) is the derivative of Elliptic[ArcSin[x]/2,2]. At the FriCASIntegration website

\begin{axiom}
setSimplifyDenomsFlag(true)
integrate(1/(1-x^2)^(1/4), x)
\end{axiom}

is unable to find a closed-form antiderivative. So apparently FriCAS concurs it is elliptic.

Albert