Re: Please help overcome problems with integration

*To*: mathgroup at smc.vnet.net*Subject*: [mg72448] Re: Please help overcome problems with integration*From*: "David W. Cantrell" <DWCantrell at sigmaxi.net>*Date*: Thu, 28 Dec 2006 05:15:55 -0500 (EST)*Organization*: NewsReader.Com Subscriber*References*: <200612151206.HAA00084@smc.vnet.net> <em39l4$7bb$1@smc.vnet.net> <emb59i$9mi$1@smc.vnet.net>

"dimitris" <dimmechan at yahoo.com> wrote: > Maybe the following link will help you a little > > http://www.cs.cmu.edu/~adamchik/articles/mier.htm The first part of that article addresses the problem, yes. But some comments about that part of the article seem in order. Adamchik begins by mentioning Integrate[(x^2 + 2x + 4)/(x^4 - 7x^2 + 2x + 17), {x, 0, 4}] and then notes that the corresponding indefinite integral int = Integrate[(x^2 + 2x + 4)/(x^4 - 7x^2 + 2x + 17), x] can be expressed in terms of elementary functions. Simplified (which was not done in his article), we get int = ArcTan[(1 + x)/(4 - x^2)]. That antiderivative has jump discontinuities at x = +/- 2 and so, of course, we cannot _naively_ use it with the Fundamental Theorem to evaluate the definite integral previously mentioned. Adamchik then shows how int can be used with the Fundamental Theorem to evaluate the definite integral, namely, Limit[int, x -> 4, Direction -> 1] - Limit[int, x -> 2, Direction -> -1] + Limit[int, x -> 2, Direction -> 1] - Limit[int, x -> 0, Direction -> -1] and states that "Mathematica evaluates definite integrals in precisely that way." [I was quite surprised by that statement. Is that always the case?] A little later, he says "The origin of discontinuities along the path of integration is not in the method of indefinite integration but rather in the integrand." If that were true, since the integrand cannot be changed, it would seem that there would be absolutely no way to avoid discontinuities along the path of integration. But that is simply not the case! Here, for example, is an antiderivative which is _continuous along the whole real line_: In[37]:= contint = ArcTan[(4x + Sqrt[2(15 + Sqrt[241])])/(2 - Sqrt[2(-15 + Sqrt[241])])] + ArcTan[(4x - Sqrt[2(15 + Sqrt[241])])/(2 + Sqrt[2(-15 + Sqrt[241])])]; While it might be difficult to use Mathematica to find contint, it is at least easy to use Mathematica to confirm that it is an antiderivative: In[38]:= Simplify[D[contint, x]] Out[38]= (4 + 2*x + x^2)/(17 + 2*x - 7*x^2 + x^4) Adamchik mentions next that the four zeros of the integrand's denominator are two complex-conjugate pairs having real parts +/- 1.95334. It then seems that he is saying that, connecting these conjugate pairs by vertical line segments in the complex plane, we get two branch cuts... But didn't the relevant branch cuts for his int cross the real axis at x = +/- 2, rather than at x = +/- 1.95334? (Note: The difference between 1.95334 and 2 is not due to numerical error.) Exactly what's going on here? Later on, he says "Thinking hard, we can build an antiderivative that does not have a branch cut crossing a given interval of integration. However, we can never get rid of branch cuts!" I agree. However, integration along the real line is often so useful that it would be nice to be able to ask Mathematica, as an option, to give an antiderivative (such as my contint above) which is continuous along the real line. In the last part of that section, Adamchik notes that "...Integrate may not be able to detect all singular points on the interval of integration, which will result in a warning message..." But it's also worth noting that such a failure may occur _without_ any warning message. A recent example was posted to this newsgroup by Dimitris in the thread "integrate"; the problem with his last item seems to be caused by branch cuts of EllipticF[z, m]. David W. Cantrell

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