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Re: An independent integration test suite
Posted:
Apr 25, 2013 12:30 PM
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clicliclic@freenet.de wrote:
>
> Waldek Hebisch schrieb:
> >
> > clicliclic@freenet.de wrote:
> > >
> > > It has occured to me that Waldek should post the FriCAS results for the
> > > two cases he could not verify by differentiation. Perhaps they are
> > > correct, perhaps not. I doubt that results with mixed branches should be
> > > counted as successes: numerical evaluation would fail even on FriCAS, I
> > > suppose.
> > >
> >
> > (1) -> integrate((asin(x/a)*asin(x/a)^(1/2))/((-1*x^2+a^2)^(1/2)), x)
> >
> > +-------------------+
> > x 2 | x
> > 2atan(------------) |2atan(------------)
> > +---------+ | +---------+
> > | 2 2 | | 2 2
> > \|- x + a \| \|- x + a
> > (1) ------------------------------------------
> > +-+
> > 5\|2
> > Type: Union(Expression(Integer),...)
> > (2) -> integrate(acos((x/(x+1))^(1/2)), x)
> >
> > +-----+
> > 1 | x +-----+
> > (2) (x + 1)atan(----------------) + |----- \|x + 1
> > +-----+ \|x + 1
> > | x +-----+
> > |----- \|x + 1
> > \|x + 1
> > Type: Union(Expression(Integer),...)
> >
>
> These are correct antiderivatives for integrands
>
> INT(ATAN(x/SQRT(a^2 - x^2))^(3/2)/SQRT(a^2 - x^2), x) =
> 2/5*ATAN(x/SQRT(a^2 - x^2))^(5/2)
>
> INT(ATAN(1/(SQRT(x/(x+1))*SQRT(x+1))), x) =
> (x+1)*ATAN(1/(SQRT(x/(x+1))*SQRT(x+1))) + SQRT(x/(x+1))*SQRT(x+1)
>
> that differ from the original ones in the omission of piecewise constant
> prefactors. I guess such a rewriting of integrands in FriCAS could take
> place as a preparatory step or, perhaps more likely, implicitly during
> their passage through the Risch mill.
Note that as real functions asin(x/a) and atan(x/sqrt(a^2 - x^2)) are
defined on the same domain (that is for a > 0 interval [-a, a]) and
take the same values.
The same for acos((x/(x+1))^(1/2)) and atan((x/(x+1))^(1/2)*(x+1)^(1/2)).
FriCAS assumes that functions to integrate are real and ATM uniformly
performs such changes.
You of course may have your own opinion, but IME for complex
integration paths one usualy wants to choose version which
is analytic along the path (if possible). This usually
requires separate resoning.
Note that if you consider functions as defined in complex plane,
but having branch cuts, than antiderivatives are incorrect for
simple fact that discontinous functions have nontrivial
distributional part of derivative.
--
Waldek Hebisch
hebisch@math.uni.wroc.pl
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