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MathGroup Archive 2004

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Re: One question about Diracdelta function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg46959] Re: One question about Diracdelta function
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Wed, 17 Mar 2004 02:29:10 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <c388np$bo0$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <c388np$bo0$1 at smc.vnet.net>,
 Daohua  Song <ds2081 at columbia.edu> wrote:

> hey,Group
>     i want to do an integration like
>     (N)Integrate[Diracdelta[f[x,y]-a],{x,min.max},{y,min,max}]
>     a is a constant number, f[] is a funtion in the form of cos,sin
>     It seems impossible to get exact answer by using Intergate command, 

Well, it is non-trivial in general. Try the following example to see why:

   Integrate[DiracDelta[Cos[x] - a], {x, b, c}]

> So is it possible to get an answer with Numerical Nintegrate?

NIntegrate will not help. You may be able to do the integration exactly 
for numerical parameters. For example,

  Integrate[DiracDelta[Sin[x y] - 1/2], {x, 1, 2}, {y, 0, 1}]

(You get an inverse function warning message if you try this).

>     another trivial question is: Abs[Exp[ikx]....]^2,
>     How can i tell the mathmatica to express it in form of Cos, Sin?

Use ComplexExpand (possibly with TargetFunctions -> {Re, Im}), e.g.,

  ComplexExpand[Exp[I k x], TargetFunctions -> {Re, Im}]

Note that Conjugate only works with numerical arguments. In most 
situations adding the definition

  SuperStar[z_] := z /. Complex[a_,b_] :> Complex[a,-b] 

(conjugation by adding a superscript * to the expression) or

  OverBar[z_] := z /. Complex[a_,b_] :> Complex[a,-b] 

(using an overbar instead) work nicely.

Cheers,
Paul

-- 
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
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Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul


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