Re: (Newbie) More confusion with integral of absolute vals
- To: mathgroup at smc.vnet.net
- Subject: [mg9422] Re: (Newbie) More confusion with integral of absolute vals
- From: Pedro A Santos <pasantos at math.ist.utl.pt>
- Date: Wed, 5 Nov 1997 01:56:38 -0500
- Organization: I.S.T. - Portugal
- Sender: owner-wri-mathgroup at wolfram.com
David Withoff wrote: > > I want to stop short of describing integration of absolute value as > hopeless, since problems such as these certainly can in principle be > fixed, but this is a difficult and largely unsolved problem, and the > fact that Maple and Mathematica have various difficulties integrating > absolute value is not at all surprising. It is a good idea to check > the results for integrals involving functions like absolute value no > matter what system you use, and where possible an even better idea is > to somehow rearrange the input (such as by using Sqrt[x^2] in place of > Abs[x]) so that these problems don't come up in the first place. Unfortunately, I've found more of these type of problems with Integrate, when trying to use Mathematica with step or piecewise continuous functions. For instance, Mathematica can't give the primitive Integrate[If[0<x<1,1,0],x]. The kind of rearrangements you sugest (in the above case, for instance writing UnitStep[x]-UnitStep[x-1] instead of If[0<x<1,1,0]) can be dangerous because 1- Usually the expressions are not completly equivalent (Sqrt[x^2] is equal to Abs[x] only for real numbers). 2- Even if they are, some expressions when used in the internal algoritms of Mathematica can lead to other types of problems (UnitStep is a function with unbounded domain and infinite integral, for example). As it is, Mathematica has three different ways to represent piecewise continuous functions (Conditionals, UnitStep, Signal), but the internal treatment is different. This leads to varying responses by the same commands (Integrate, FourierTransform, etc) to what is mathematically equivalent, and is very anoying when I am trying to work with Mathematica. Is there a solution for this problem? Pedro Santos