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Re: DiracDelta Integrals
*To*: mathgroup at smc.vnet.net
*Subject*: [mg9356] Re: DiracDelta Integrals
*From*: "Stephen P Luttrell" <luttrell at signal.dra.hmg.gb>
*Date*: Sat, 1 Nov 1997 03:33:42 -0500
*Organization*: Defence Evaluation and Research Agency
*Sender*: owner-wri-mathgroup at wolfram.com
> just out of curiosity, i tried
>
> Integrate[ Exp[I k x], {x, -Infinity, Infinity}]
>
> and Mathematica said "the integral is 0 if Im[k]==0, otherwise, i give
> up". so, thinking i was being slick, i loaded the
> Calculus`DiracDelta` package.
>
>...DELETIA...
In Mathematica 3 the following input:
<<Calculus`FourierTransform`
FourierTransform[1, t, w]
Gives the following output:
2 \[Pi] DiracDelta[w]
If you want to obtain this result using Integrate, then here is a rather
indirect
way of doing it.
The following inputs:
b =Integrate[Exp[I w t-a Abs[t]],{t,-Infinity,Infinity},
Assumptions->{Im[w]==0,Re[a]>0}] b/.{a->0}
Integrate[b,{w,-w0,+w0}]//Limit[#,a->0]&//PowerExpand
produces the following outputs:
\!\(\(2\ a\)\/\(a\^2 + w\^2\)\)
0
2 \[Pi]
Here I use the parameter "a" as a regulariser, which ensures that the
integral converges
as Abs[t]->Infinity. The integral has poles at w==+I a and w==-I a,
which pinch the integration contour
as a->0. For w^2>0 the output from second line of the input shows that
the integral is 0, whereas the
output from third line of the input shows that the integral (from w==-w0
to w==+w0) of the integral
is 2 \[Pi].
That demonstrates that the integral is 2 \[Pi] DiracDelta[w], as
required.
----------------------------------------------------------------------------
---------------------------------------- Stephen P Luttrell
luttrell at signal.dra.hmg.gb
Adaptive Systems Theory 01684-894046 (phone)
Room EX21, DERA 01684-894384 (fax)
Malvern, Worcs, WR14 3PS, U.K.
http://www.dra.hmg.gb/cis5pip/Welcome.html
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