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Re: Integrating DircaDelta[x]

  • To: mathgroup at smc.vnet.net
  • Subject: [mg78528] Re: Integrating DircaDelta[x]
  • From: dimitris <dimmechan at yahoo.com>
  • Date: Tue, 3 Jul 2007 05:43:12 -0400 (EDT)
  • References: <f6al5g$hgd$1@smc.vnet.net>

            Pillsy       :
> In Mathematica 6, integrating the DiracDelta function with specified
> limits gives the expected result:
>
> In[1]:= Integrate[DiracDelta[x], {x, -Infinity, -1}]
>
> Out[1]:= 0
>
> In[2]:= Integrate[DiracDelta[x], {x, -Infinity, 1}]
>
> Out[2]:= 1
>
> In[3]:= Integrate[DiracDelta[x], {x, -Infinity, 0}]
>
> Out[3]:= 1/2
>
> But when you replace the limit with a variable, it returns something
> quite different:
>
> In[4]:= Integrate[DiracDelta[x], {x, -Infinity, a}]
>
> Out[4]:= If[a \[Element] Reals, 1,
>  Integrate[DiracDelta[x], {x, -\[Infinity], a},
>   Assumptions -> Im[a] < 0 || Im[a] > 0]]
>
> In[5]:= Map[% /. a -> # &, {-1, 0, 1}]
>
> Out[5]:= {1, 1, 1}
>
> Any idea what's going on? I'm using the 32-bit x86 Mac version on OS X
> 10.4.10, if it matters.
>
> TIA,
> Pillsy

Hi.

$VersionNumber->5.2

In[44]:=
Integrate[DiracDelta[x], {x, -Infinity, -1}]
Out[44]=
0

In[45]:=
Integrate[DiracDelta[x], {x, -Infinity, 1}]
Out[45]=
1

In[46]:=
Integrate[DiracDelta[x], {x, -Infinity, 0}]
Out[46]=
1/2

In[47]:=
Integrate[DiracDelta[x], {x, -Infinity, a}]
Out[47]=
UnitStep[a]

In[48]:=
PiecewiseExpand[%]
Out[48]=
Piecewise[{{1, a >= 0}}]

However, e g

In[54]:=
Integrate[DiracDelta[x], {x, -Infinity, 2*I}]
Out[54]=
Integrate[DiracDelta[x], {x, -Infinity, 2*I}]

So Mathematica 6 does a better job since it
returns an If structure.

Nevertheless, as you point out neither 6 is perfect.

Using the function PiecewiseIntegrate of Maxim Rytin
available from here (http://library.wolfram.com/infocenter/MathSource/
5117/)
we get

In[116]:=
PiecewiseIntegrate[DiracDelta[x], {x, -Infinity, a}]
Out[116]=
If[0 <= a, 1, 0]

Best Regards
Dimitris



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