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?
--
Michel Talon