
Re: can your system handle 4throot pseudoelliptics?
Posted:
Sep 23, 2017 4:41 PM


On Saturday, September 23, 2017 at 7:37:49 AM UTC10, clicl...@freenet.de wrote: > Albert Rich schrieb: > > > > On Friday, September 22, 2017 at 6:58:04 AM UTC10, 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 pseudoelliptic. 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/((2x^2)*(x^21)^(1/4)) Rubi 4.13.3 gets > > > > ArcTan[x/(Sqrt[2]*(x^21)^(1/4))]/Sqrt[2] + > > ArcTanh[x/(Sqrt[2]*(x^21)^(1/4))]/Sqrt[2]  > > 2*(1x^2)^(1/4)/(x^21)^(1/4)*EllipticE[ArcSin[x]/2,2] > > > > involving only a single EllipticE function. If this was actually a > > seudoelliptic 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 online proof > can be ordered at: > <http://axiomwiki.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 lowercase sqrt(), sin(), etc.
According to Mathematica, 1/(2*(1x^2)^(1/4)) is the derivative of Elliptic[ArcSin[x]/2,2]. At the FriCASIntegration website
\begin{axiom} setSimplifyDenomsFlag(true) integrate(1/(1x^2)^(1/4), x) \end{axiom}
is unable to find a closedform antiderivative. So apparently FriCAS concurs it is elliptic.
Albert

