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Integrate?
*To*: "mathgroup" <mathgroup at yoda.ncsa.uiuc.edu>
*Subject*: Integrate?
*From*: "WILLIAM M. GOLDING" <golding at ccf1.nrl.navy.mil>
*Date*: 5 Nov 90 16:07:00 EDT
We've seen that Mathematica gives incorrectly a result of zero
for this one particular integral,
Integrate[Sqrt[2-2 Cos[x]],{x,0,2 Pi}] .
Because the Sqrt function has a branch cut along the negative real axis,
and it seems likely that Mathematica would try to do the trigonometric
integral in the complex plane thereby encountering the cut at x or "theta"
equal to Pi, I translated the integrand from x = Pi to x = 0 and
integrated from -Pi to Pi thinking that this might make it easier for mma
to deal with the cut. That is I tried the integral
Integrate[Sqrt[2+2 Cos[x]],{x,-Pi, Pi}] .
This integral is also returned incorrectly as zero. The second integrand is
obtained from the first by letting x go to x - Pi.
Next it seemed reasonable to look at the two integrals over the same
region of integration. That is
Integrate[Sqrt[2-2 Cos[x]],{x,0, Pi}] => 4
Integrate[Sqrt[2+2 Cos[x]],{x,0, Pi}] => - 4
The two integrals should be identical. The only difference between the
two integrands is that the first is monotonically increasing and the
second is monotonically decreasing. Is this the cause of the sign error?
It seems that mma is taking perfectly good symmetric functions and making
them antisymmetric by multiplying by +1 if going uphill and multiplying
by -1 if going downhill.
There is another integral which shows a similar behaviour ,
Integrate[ Sqrt[ Sec[x]-1 ],{x,0, Pi/2}] => 1.76275
Integrate[ Sqrt[ Sec[x]-1 ],{x,- Pi/2,0}] => - 1.76275
That is we get a positive result on the uphill side and a negative result on
the downhill side. In each of these integrals, if you use the second order
series expansion of the trigonometric function in place of the trig function,
this sign flipping phenomenon appears to go away and for a small range chosen
symmetrically about the symmetry axis the integrate function returns a correct
non-zero result. This seems to imply that mma will likely have trouble of the
of the above sort with integrands of the form
Sqrt[ Function[ any trig function ]]
where Function is non-negative in the range of integration
and where the integration is taken through a complete period of the trig
functions involved.
I would like to hear of any functions in the above form that mma
can handle correctly, and also if anyone can explain why Mathematica
should have problems with integrals of the above form.
Thanks
Mike Golding
Email golding at ccf.nrl.navy.mil
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