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