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Re: Re: Integrating DiracDelta to get UnitStep

• To: mathgroup at smc.vnet.net
• Subject: [mg91293] Re: [mg91261] Re: Integrating DiracDelta to get UnitStep
• From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
• Date: Wed, 13 Aug 2008 04:42:39 -0400 (EDT)
• References: <g7p2tm$arr$1@smc.vnet.net> <200808120846.EAA18776@smc.vnet.net>

On 12 Aug 2008, at 10:46, David W.Cantrell wrote:

> 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
>


Also,

Integrate[DiracDelta[x], {x, -Infinity, t},
    Assumptions -> t < 0]
0

Integrate[DiracDelta[x], {x, -Infinity, t},
    Assumptions -> t <= 0]
  HeavisideTheta[t]

Integrate[DiracDelta[x], {x, -Infinity, t},
    Assumptions -> t > 0]
1

all of which are correct.

Andrzej Kozlowski

• References:
  • Re: Integrating DiracDelta to get UnitStep
    • From: "David W.Cantrell" <DWCantrell@sigmaxi.net>