This is a typical ill-posed problem resulting from the limited mind-set
that comes from
thinking that sqrt is a single-valued function, and that therefore
sqrt(z^2), an expression which can be reduced to a set: {-z,z} should be
collapsed to abs(z).
Which it of course is not. Draw the graphs of f(z)=-z or f(z)=z, and
you see
that NEITHER ONE is the same as f(z)=abs(z).
So, given the very strong possibility that the limited mind set occurs in
at least one of 3 places:
the mind of the proposer
the mind of the Sage programmer
the mind of the Maxima programmer
what is one to do?
Do you want to integrate cot(z), which indeed gives an integral of 0,
correctly.
or do you want to integrate -cot(z), which of course also gives 0
or abs(cot(z)) which would be integral(cot(z),z,%pi/4,%pi.2) -
integral(cot(z),z,%pi/2,3*%pi/4).
and that gives log(2).
in the Maxima I have on my computer, integrate(abs(cot(z)),z,%pi/4,
3*%pi/4) gives the very
peculiar answer log(-1)
the numerical integration program quad_qag gives 0.69314718055995 which
looks like log(2).
So there is probably no bug in the integration version of the answer that
is 0, There is a bug
in sqrt. Or rather, in the mind-set regarding sqrt.
There is a bug in the integration of abs(cot()) though. probably related to
Nils' observation, though
probably not in the way he thinks... the integral is of a real function
between real values.
Also if Sage translates a definite integration to a call to Maxima for an
INdefinite integration,
and then using the fundamental theorem of calculus (difference of value of
integral at endpoints)
it is making a BIG MISTAKE. FTC doesn't always apply, and Maxima knows
something about
definite integrals, contour integration etc. Not apparently bug-free..
RJF
On Thursday, August 27, 2015 at 10:10:39 AM UTC-7, Gregory Bard wrote:
>
> There is an integral which Sage correctly numerically integrates, and
> which Sage symbolically gets very wrong. William and I looked into this
> during Sage Days 68, and he discovered that, in fact, Maxima gets this
> integral very wrong as well. (More correctly, the particular configuration
> and version of Maxima built into Sage gets the integral very wrong. Some
> setting might be incorrect or badly chosen.)
>
> The integral of sqrt( cot(x)^2 ) dx for pi/4 < x < 3pi/4 is not too bad to
> compute by hand. The answer is the logarithm, base e, of 2. The numerical
> integration in Sage agrees.
>
> Maxima thinks that the answer is instead zero. (Or more correctly, the
> particular configuration and version of Maxima built into Sage thinks the
> answer is zero.)
>
> Please see the attached sagews worksheet.
>
> Is there anyone on this list who knows Maxima in general, and the
> connections/interface between Maxima and Sage in particular?
> ---Greg
>
>
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