To some extent these complaints about integration are pointing out
problems that were,
I suspect, not part of the original formulation.
The Risch algorithm finds antiderivatives of functions consisting of the
composition of various
elementary and not-so-elementary functions. It is an algebraic
algorithm and does not
concern itself with concepts like continuity. Using the Risch algorithm
+ a trivial application
of the fundamental theorem of integral calculus ... F(upper) -
F(lower) is basically a
dangerous way to do definite integrals, and so there is a different set
of programs for definite
integration.
The definite integration package, I suspect is also being misapplied. I
think it is intended to
find real valued integrals of real valued functions on the real line.
To do so it uses techniques
derived from contour integration, but it does not really serve the role
of allowing the user to
define a contour. Though that can be concocted by some torturous
transformations ...
Maybe one way to deal with these issues is to check the input and when
possible, barf early rather than later.
RJF
On 11/23/2011 8:07 AM, Michel Talon wrote:
> While looking for information on maxima stuff, i fall on a message from
> Fateman
> http://www.math.utexas.edu/pipermail/maxima/2001/000479.html
> so, ten years ago, who complains that several CAS give false answers for the
> integral
> integrate(1/sqrt(2-2*cos(x)),x,-%pi/2,%pi/2)
> I have just tried it on maxima-5.25.1 and it gives:
> Principal Value
> (%o3) 0
> This is one more problematic result compared to Fateman's list!
> We have the indication that a principal value occurs somewhere. In fact
> in the middle of the integration range we have a problem at x=0 where
> 2-2*cos(x) ~ x^2 and thus we have 1/sqrt(1/x^2) ~ 1/x
> Which is a divergent integral, as Fateman says. If we postulate that sqrt(x)>0
> for x>0, the integral is clearly divergent. If we model the integrand to 1/x
> around x=0, that is, we choose a change of sign when crossing 0, then this is
> coherent with a principal value, and indeed the function becomes antisymmetric
> in x-> -x and the integral is rightly 0. One more problem related to the
> multivaluedness of square roots.
> What do you think is the most correct or most convenient way to treat this
> integral in maxima?
>
>