[Axiom-developer] [Axiom-mail] Non elementary integration

[Martin Rubey]

Changes http://page.axiom-developer.org/zope/mathaction/AxiomMail/diff
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Dear Stefano,

Stefano Simonucci writes:

 > Now if I write
 > integrate(exp(x)/x,x)
 > I obtain
 > Ei(x)
 > while if I write
 > integrate(exp(x)/x^2,x)

 > I get a form integral. But the integral of exp(x)/x^2 can be given in terms
 > of Ei(x) as exp(x).  In fact I believed that the axiom can be able to find
 > the solution that is
 
 > integrate(exp(x)/x^2,x) --> Ei(x)-exp(x)/x

this would be correct, yes.

 > But from the manual I deduce that exp(x)/x^2 can be prooved not elementary
 > integrable. Why exp(x)/x is integrable while exp(x)/x^2 not?

Well, exp(x)/x is *not* elementarily integrable, Ei(x) is simply defined to be
its integral. Ei(x) is *not* an elementary function. (in the mathematical
sense)

However, Axiom should still be able to figure out your integral. The
corresponding code is in intpm.spad, but your integral doesn't fit, it seems.

If you are willing to do some research, we might be able to figure out how to
improve the pattern matcher. There is also code by Manuel Bronstein (written in
maple), that would handle special functions. Maybe this should be integrated,
too.

Note that there are also two bugs (with fixes) filed on IssueTracker, so if you
rely heavily on integrals, you might want to patch your axiom:

http://page.axiom-developer.org/zope/mathaction/198IntegrateSinX2XIsNotHandled

http://page.axiom-developer.org/zope/mathaction/199IntegrateExpX2ExpXXX

The first patch is hidden in a comment, you'll have to look for it.

Martin



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