       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)
• 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:=
> D[UnitStep[x],{x,2}]
> Integrate[%,{x,-1,1}]
> Out=
> Derivative[DiracDelta][x]
> Out=
> 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

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[DiracDelta][x]*f[x],
{x, -Infinity, Infinity}]

-Derivative[f]

Integrate[Derivative[DiracDelta][x]*f[x],
{x, -Infinity, Infinity}]

Derivative[f]

So taking f to be the function 1 we get, correctly

Integrate[Derivative[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

```

• Prev by Date: RE: Two questions (1) Sollve and (2) Precision
• Next by Date: RE: Plot Sequencies of Complex Functions
• Previous by thread: Re: piecewise integration
• Next by thread: Re: Re: piecewise integration