> > It seems that polynomials with even power in the > > leading monomial produce some issues. However > > all of these failures have very reasonable results. > > Maybe they are out of the scope of Risch for some > > reason? > > Well, at first I was surprised how much FriCAS can do...
> The 'exp(-x^2)' function is a "canonical" example of function > having no elementary integral. Yes, right. > Using integration by parts it is easy to see that integral > of 'x^(2*k)*exp(-x^2)' is a constant times integral of > 'exp(-x^2)' plus elementary part, so in this case also > there is no elementary integral. > > In case of 'x^(2*k+1)*exp(-x^2)' using integration by parts > one can easily find elementary integral. So Risch algorithm > handles odd powers with no problems, while even powers have > no elemetary integral so Risch gives up. I see, with odd exponent we hit the base case int(x*exp(-x^2)) which we can solve while for even exponent we find int(exp(-x^2)). This is fine so far concerning the algorithmic part. > It is interesting that Risch still can do orthogonal cases (with > proper weight). AFAICS this is because nonelementary parts cancel. Hmm, do they? The polynomial H_m*H_n has either only odd or only even monomials. Odd cases should be no problem at all, as we found above. > In other words we have sum where each term separately have no > elementary integral but the sum does. > In m = n case the main contribution comes from nonelementary > part and Risch can not do this. > In case m = n = 0 the integral is > handled by pattern matcher, but the case m = n > 0 is not > handled by pattern matcher. Ah well. In that case we have a polynomial times the exponential. Maybe one could extend the pattern matcher for this case, maybe not. (Another Idea would be to use the Meijer-G Ansatz to integration.) Thanks for this explanation. I see now why I get the results I got. I'd have to work out all the cases on paper now for more insight. -- You received this message because you are subscribed to the Google Groups "FriCAS - computer algebra system" group. To post to this group, send email to [email protected]. To unsubscribe from this group, send email to [email protected]. For more options, visit this group at http://groups.google.com/group/fricas-devel?hl=en.
