Re: Re: Re: (x-y) DiracDelta[x-y] does not simplify to 0

*To*: mathgroup at smc.vnet.net*Subject*: [mg56380] Re: [mg56361] Re: [mg56297] Re: (x-y) DiracDelta[x-y] does not simplify to 0*From*: Pratik Desai <pdesai1 at umbc.edu>*Date*: Sat, 23 Apr 2005 01:16:12 -0400 (EDT)*References*: <d42kg5$39t$1@smc.vnet.net> <d45agf$ieu$1@smc.vnet.net> <200504210936.FAA05048@smc.vnet.net> <0e5459acc0a6eae9a16bda863b79434c@mimuw.edu.pl> <16999.49105.639076.187149@localhost.localdomain> <913e334e36e5b09dce4f89131131357d@mimuw.edu.pl> <17000.4781.157008.559505@localhost.localdomain> <200504221024.GAA18888@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

Andrzej Kozlowski wrote: >On 22 Apr 2005, at 05:53, Alain Cochard wrote: > > > >>*This message was transferred with a trial version of CommuniGate(tm) >>Pro* >>Andrzej Kozlowski writes: >> >> >> >>>Since [use inside Integrate] is the only context in which [x >>>DiracDelta[x] == 0] is useful and makes sense, I can't see any >>>justification for your application of Simplify. >>> >>> >>OK, I think I now understand what you say and I disagree with your >>conclusion. I indeed do find practical justifications for the use of >>that identity and had so far to perform the simplifications manually. >> >> >> >> > >Of course I did not mean to doubt that this identity or this particular >way of treating the DiracDelta can't be useful for you or others. There >is no end to unexpected and ingenious uses that people make of various >built in functions in Mathematica. However, unless you actually >integrate your expressions containing the DiracDelta I do not think you >are really making use of the mathematical notion of a distribution. For >example, I could define a function MyDiscreteDelta by > >MyDiscreteDelta[x_] := DiracDelta[x]/DiracDelta[0] > > >and then I would have > > >FullSimplify[x*MyDiscreteDelta[x]] > >0 > >(Note that, as i pointed out earlier, the >FullSimplify[x*DiscreteDelta[x]] does not return 0). So one could argue >that I "used" the DiracDelta to get a "superior" version of the the >DiscreteDelta, and this may be further useful for something else. Even >if this were the case it would still be true that the mathematical >notion of a distribution is not being used here at all because that >notion is only meaningful when used in th context of integration. > >Andrzej > > > > I agree with Andrzej Kozlowski, another way to look at the Dirac Delta is that it represents a charge density for a unit charge placed at x=y, where as the less eggregious (at least to the mathematician) Heaviside Function Integrate[DiracDelta[x-y],{x,0,1}] represents the corresponding cumulative charge distribution. So the upshot is that the DiracDelta is a symbolic function, whereas the Heaviside function is a respectable piecewise continuous function. -- Pratik Desai Graduate Student UMBC Department of Mechanical Engineering Phone: 410 455 8134

**References**:**Re: (x-y) DiracDelta[x-y] does not simplify to 0***From:*Alain Cochard <alain@geophysik.uni-muenchen.de>

**Re: Re: (x-y) DiracDelta[x-y] does not simplify to 0***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>