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MathGroup Archive 2007

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Re: Integral of Piecewise function involving DiracDelta

  • To: mathgroup at smc.vnet.net
  • Subject: [mg74800] Re: Integral of Piecewise function involving DiracDelta
  • From: dh <dh at metrohm.ch>
  • Date: Thu, 5 Apr 2007 04:17:28 -0400 (EDT)
  • References: <euvmuo$epv$1@smc.vnet.net>


Hi Andrew,

Obviously there is a bug in the implementation of Pieceweise or its 

integral. Also note that: 

Integrate[Piecewise[{{DiracDelta[x],-1<x<1}},0],{x,-.1,.1}] evaluates to 1.

Further, concerning  your second question. It is well known that 

DiracDelta[a x] == DiracDelta[x] / a for a constant a>0. A handwaving 

argument is, that the dirac function becomes "narrower" by a factor of 

a, what makes the integral smaller by the same factor.

Daniel



Andrew Moylan wrote:

> Here is an integral that I expect Mathematica to evaluate to 1:

> 

> Integrate[Piecewise[{{DiracDelta[x], -1 < x < 1}}, 0],

>   {x, -Infinity, Infinity}]

> 

> However, Mathematica 5.2 (Windows) gives the answer as 0. Here's a

> similar integral that I also expect to evaluate to 1:

> 

> Integrate[Piecewise[{{DiracDelta[x-1/2], -1 < x < 1}}, 0],

>   {x, -Infinity, Infinity}]

> 

> For this integral, Mathematica doesn't return 0. It returns the

> following:

> 

> Integrate[Piecewise[{{2*DiracDelta[-1 + 2*x], -1 < x < 1}},

>   0], {x, -Infinity, Infinity}]

> 

> Can anyone help me understand what's happening here?

> 

> Cheers,

> 

> Andrew

> 

> 




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