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Re: Re: piecewise integration


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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