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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)
*References*: <20060605102611.774$dR_-_@newsreader.com> <200606061028.GAA20748@smc.vnet.net> <acbec1a40606071502r7aea9e4ahcea554c39976f739@mail.gmail.com> <B4EA9689-6BEF-4F01-AD21-FC9C6EAFC212@mimuw.edu.pl> <acbec1a40606072149na61bec9ofb16ce601628afbb@mail.gmail.com> <acbec1a40606072200i1339bcf2jbd43808da9204a9c@mail.gmail.com>
*Sender*: owner-wri-mathgroup at wolfram.com
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
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