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Re: Integrating DiracDelta to get UnitStep

• To: mathgroup at smc.vnet.net
• Subject: [mg91261] Re: Integrating DiracDelta to get UnitStep
• From: "David W.Cantrell" <DWCantrell at sigmaxi.net>
• Date: Tue, 12 Aug 2008 04:46:25 -0400 (EDT)
• References: <g7p2tm$arr$1@smc.vnet.net>

CRC <crobc at this-is-bogus.sbcglobal.net> wrote:
> Hi:
>
> I am a bit confused by Mathematica 6.0.3 behavior.  I expect that:
>
> In[n]:= Integrate[DiracDelta[x], {x, -\[Infinity], t},
>   Assumptions -> Im[t] == 0]
>
> Will produce:
>
> Out[n]= UnitStep[t]
>
> But instead it produces:
>
> Out[n]= 1
>
> However,
>
> In[n+1]:= Plot[ Integrate[DiracDelta[x], {x, -\[Infinity], t},
>    Assumptions -> Im[t] == 0], {t, -2, 2} ]
>
> produces the expected plot of UnitStep[t].
>
> Why doesn't the integration output the UnitStep function?

I share your concern and will be interested to see what others say.
Here's something which is closely related but which works as it should:

In[2]:= Integrate[DiracDelta[x], {x, t, Infinity},
 Assumptions -> Im[t] == 0]

Out[2]= HeavisideTheta[-t]

Perhaps that will help.
David

• Follow-Ups:
  • Re: Re: Integrating DiracDelta to get UnitStep
    • From: Andrzej Kozlowski <akoz@mimuw.edu.pl>