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