Re: Re: Apparent error integrating product of DiracDelta's

*To*: mathgroup at smc.vnet.net*Subject*: [mg92098] Re: [mg92073] Re: Apparent error integrating product of DiracDelta's*From*: Daniel Lichtblau <danl at wolfram.com>*Date*: Sat, 20 Sep 2008 04:57:18 -0400 (EDT)*References*: <gag2lg$39k$1@smc.vnet.net> <gal3ht$dv1$1@smc.vnet.net> <200809190916.FAA15073@smc.vnet.net>

magma wrote: > On Sep 18, 12:10 pm, Daniel Lichtblau <d... at wolfram.com> wrote: >> magma wrote: >>> On Sep 15, 9:40 am, "Nasser Abbasi" <n... at 12000.org> wrote: >>>> "Michael Mandelberg" <mmandelb... at comcast.net> wrote in message >>>> news:gag2lg$39k$1 at smc.vnet.net... >>>>> How do I get: >>>>> Integrate[DiracDelta[z- x] DiracDelta[z- y], {z-Infinity, Infinit= > y}= >>> ] >>>>> to give DiracDelta[x-y] as the result? Currently it gives 0. I = > ha= >>> ve >>>>> all three variable assumed to be Reals. I am using 6.0.0. >>>>> Thanks, >>>>> Michael Mandelberg [...snip my claim to the effect that the result is not defined at x==y] > Very interesting answer! Yes. I have to hope it was not so interesting as to have gone to the point of fiction. > Yet, taking a naive (physicist's ?) point of view, the result seems > intuitively correct. > Mr. Delta himself (P.A.M. Dirac) in his "The principles of quantum > mechanics" published in 1930 confirms the result (eq. 15.9 in the > Italian 4th edition). > He even "proves" it, a little bit later. The "proof" is a bit > heuristics because he assumes the integral is defined in the first > place. > Yet the whole Delta function story was at the time just heuristics. > Apparently the delta was described for the first time in this book > (Wikipedia). > I am also pretty sure he uses the result somewhere else later in the > book (probably where double integrals appear). > So, could he have been so wrong after having been so right before? If it's a question of Dirac being wrong, or me, I'd put the money on me being wrong. That said, I do not know precisely what it was he claimed, or did, in this particular setting. It might be he used a shaky justification for an end result that was correct. > More to the point: I found very intriguing your claim that " If > different methods of approximation will lead to different results, > then what you have cannot be defined". > I agree with you of course, but could you "show" with Mathematica - maybe with > a Manipulate program - that this is the case with this integral? Illustrating pathologies with singular integrals is not one of my stronger points (which may be why I seem to get tangled in them). I need to give this some thought. > Could you find 2 different approximation methods which give 2 > different results? > I look forward to seeing it. > > Concerning DiracDelta[x]^2, I would agree it is undefined at least > because its integral, if it existed, would be DeltaDirac[0], which is > undefined. Daniel Lichtblau Wolfram Research

**References**:**Re: Apparent error integrating product of DiracDelta's***From:*magma <maderri2@gmail.com>