Re: Re: piecewise integration
- To: mathgroup at smc.vnet.net
- Subject: [mg67070] Re: [mg66999] Re: piecewise integration
- From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
- Date: Thu, 8 Jun 2006 04:54:34 -0400 (EDT)
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On 8 Jun 2006, at 14:00, Chris Chiasson wrote:
> And on a related note, does anyone know why Mathematica handles
> DiracDelta'[x] in this way:
>
> In[1]:=
> D[UnitStep[x],{x,2}]
> Integrate[%,{x,-1,1}]
> Out[1]=
> Derivative[1][DiracDelta][x]
> Out[2]=
> 0
I can see nothing wrong with the above.
There is the following rule for the derivative of the DiracDelta:
Integrate[Derivative[n][DiracDelta][x]*f[x], {x, -Infinity,
Infinity}] == (-1)^n f[0]
for any suitable function f (a function of "slow growth").
Mathematica knows this for every positive n, e.g.
Mathematica knows this rule for any positive integer n:
Integrate[Derivative[5][DiracDelta][x]*f[x],
{x, -Infinity, Infinity}]
-Derivative[5][f][0]
Integrate[Derivative[6][DiracDelta][x]*f[x],
{x, -Infinity, Infinity}]
Derivative[6][f][0]
So taking f to be the function 1 we get, correctly
Integrate[Derivative[1][DiracDelta][x],
{x, -Infinity, Infinity}]
0
which is as it should be. But now, what is puzzling me is this:
Assuming[Element[n,Integers]&&n>0,
Integrate[Derivative[n][DiracDelta][x]*f[x],{x,-
Infinity,Infinity}]]
0
which is obviously wrong! This is with Mathematica 5.1. I wonder if
this is still so in 5.2.
Andrzej Kozlowski
- References:
- Re: piecewise integration
- From: Andrzej Kozlowski <akoz@mimuw.edu.pl>
- Re: piecewise integration