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

**References**:**Re: piecewise integration***From:*Andrzej Kozlowski <akoz@mimuw.edu.pl>