Integrate?

*To*: mathgroup at yoda.ncsa.uiuc.edu*Subject*: Integrate?*From*: Jeffrey P. Golden <jpg at ALLEGHENY.SCRC.Symbolics.COM>*Date*: Sat, 3 Nov 90 20:55 EST

Since one of you brought MACSYMA into this, I guess I'll respond: Date: Fri, 2 Nov 90 18:21:48 +0100 From: Harald Hanche-Olsen <hanche at imf.unit.no> Robert Staerk <staerk at iam.unibe.ch> writes ... Mathematica gives a wrong answer to the following integral: 2 Pi / | 1/2 | (2 - 2 cos(x)) dx | / 0 Is this a known bug? In[1]:= Integrate[ Sqrt[2 - 2 Cos[x]], {x, 0, 2 Pi}] Out[1]= 0 MAPLE V > int( sqrt(2 - 2 * cos(x)), x=0...2*Pi ); 8 As I'll try to demonstrate below, I don't think this is such an easy problem. So, I wonder how Maple did it? This is apparently caused by Mathematicas attempts at being clever about things like complex square roots, branch cuts and the like. I don't believe this theory. As I show below, it is quite possible to get the answer of 0 by simply not being clever! I don't actually know how Mathematica gets its answer of 0, a WRI person would have to tell us that. But I do know how MACSYMA gets its answer of 0. Mathematica might be losing here in the same way. Yes, it is known. And some people think Mathematica is brain-dead because of it. Myself, I haven't decided yet. I keep alternating between Mathematica and Maple, just like you seem to do... All CAS unfortunately get some integrals wrong. I only fixed MACSYMA a few months ago to get e.g. integrate(sqrt(x+1/x-2),x,0,1); -> 4/3, instead of -4/3 . How does MAPLE V do on that one? ------------------------------------------------ Date: Thu, 1 Nov 90 13:52:13 -0500 From: Jack <jack at brahms.udel.edu> Macsyma also gives the same incorrect result 0 for this integral (version 416.48 for Sparc Computers). That really surprised me. Here's how MACSYMA gets 0: (c1) /* The following gives the wrong answer: */ integrate(sqrt(2-2*cos(x)),x,0,2*%pi); (d1) 0 (c2) /* Here's the reason why: */ /* Just look at the indefinite integral and then take limits: */ integral: integrate(sqrt(2-2*cos(x)),x); 4 (d2) - ----------------------- 2 sin (x) sqrt(------------- + 1) 2 (cos(x) + 1) (c3) upper_limit: limit(integral,x,2*%pi,minus); (d3) - 4 (c4) lower_limit: limit(integral,x,0,plus); (d4) - 4 (c5) upper_limit - lower_limit; (d5) 0 (c6) /* Two ways to get the correct answer: */ /* The first works in any MACSYMA (I think), the latter only in the development version: */ 2*integrate(sqrt(2-2*cos(x)),x,0,%pi); (d6) 8 (c7) integrate(sqrt(2-2*cos(x)),x,-%pi,%pi); (d7) 8 The above is not meant to offer any excuses for giving an incorrect result. A wrong answer is bad news. C7/D7 shows that we've done something, but not enough. There are many ways to get the correct answer here: split the integration interval (and do all the consequent integrals correctly!), do a smart change of variables, use the halfangle formula, etc. But not all of these approaches are necessarily appropriate for a computer algebra system. Nevertheless, these approaches indicate to me, time allowing, that we can fix this bug in MACSYMA. I'm not promising when. Again, bugs are bad news. But unfortunately all CAS have them. We try to do the best here we can. (I have been planning a MACSYMA Newsletter article on this problem and related problems. We need to tell our users what they unfortunately must watch out for.) Anyway, this is a Mathematica Group list, so I apologize for going on so long. From: Jeffrey P. Golden <jpg at ALLEGHENY.SCRC.Symbolics.COM> Organization: Symbolics MACSYMA Division