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Re: Folding Deltas
*To*: mathgroup at smc.vnet.net
*Subject*: [mg56940] Re: Folding Deltas
*From*: Maxim <ab_def at prontomail.com>
*Date*: Tue, 10 May 2005 03:43:16 -0400 (EDT)
*References*: <d5muhb$dqs$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
On Mon, 9 May 2005 06:04:27 +0000 (UTC), <baermic at yahoo.com> wrote:
> Can anyone help to verify in Mathematica the expression given by Rota
> (http://xoomer.virgilio.it/maurocer/Text07.htm):
>
> Convolution ( Sum of DiracDeltaFct ** Sum of DiracDeltaFct) == Sum
> (DiracDeltaFct + Values).
>
> I tied
>
> Integrate[DiracDelta[t] DiracDelta[t - 2] , {t, -3, 3} ]
> which does not evaluate;
> but
>
> Integrate[DiracDelta[t] DiracDelta[t - 2] , {t, -3, 1} ] +
> Integrate[DiracDelta[t] DiracDelta[t - 2] , {t, 1, 3} ] == 0
> True
>
> ( I use ver 5.1 with W2k)
>
The typesetting is not quite correct: a_i+b_j should be in subscript. You
can verify the identity (after writing the convolution in the proper way)
for sums with fixed number of terms using PiecewiseIntegrate from
http://library.wolfram.com/infocenter/MathSource/5117/ :
In[1]:=
PiecewiseIntegrate[
(DiracDelta[tau - a1] + DiracDelta[tau - a2])*
(DiracDelta[x - tau - b1] + DiracDelta[x - tau - b2]),
{tau, -Infinity, Infinity},
Assumptions -> a1 < a2]
Out[1]=
DiracDelta[a1 + b1 - x] + DiracDelta[a2 + b1 - x] +
DiracDelta[a1 + b2 - x] + DiracDelta[a2 + b2 - x]
which just illustrates the fact that convolving with DiracDelta[x - a] is
equivalent to shifting. Another way:
In[2]:=
% == Integrate[
(DiracDelta[tau - a1] + DiracDelta[tau - a2])*
(f[x - tau - b1] + f[x - tau - b2]),
{tau, -Infinity, Infinity}] /. f -> DiracDelta
Out[2]=
True
Some of the opinions expressed in that essay seem too radical. The point
about the poor methodology of introducing integrating factors in ODE
courses is valid, but I'm not so sure about them being useless as such:
looking for an integrating factor is one of the techniques employed by
DSolve, which suggests that it covers a reasonably wide number of cases.
But then I guess one might get into a long argument about how often the
need for some things implemented in CAS (such as giving symbolic solutions
of ODEs as huge multi-page expressions involving special functions or
computing values of those special functions to thousands of digits)
actually arises in practical applications.
Maxim Rytin
m.r at inbox.ru
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