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Re: Re: delta function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg89950] Re: [mg89909] Re: delta function
  • From: Daniel Lichtblau <danl at wolfram.com>
  • Date: Wed, 25 Jun 2008 06:25:29 -0400 (EDT)
  • References: <g3ihpo$fdt$1@smc.vnet.net> <200806240729.DAA10940@smc.vnet.net>

Magician wrote:
> On Jun 21, 4:31 am, Magician <jadoo.d... at gmail.com> wrote:
>> I am integrating over a function (not written in mathematica syntax)
>>
>> u=F(x)   e^(-  (x-xo)^2/t  ) /t  ,
>> i know in the limit t ->0,  e^(-  (x-xo)^2/t  ) /t = delta(x-xo), but
>> how do i get mathematica to recognize this.
>>
>> In mathematica, how can i construct hings like the Sokhotskyi-Plemelj
>> formula ?
> 
> hey somebody plz comment????

Here are a few comments.

(1) Use Mathematica syntax to denote Mathematica expressions.
(2) Use correct formulations. Your denominator is not correct, if what 
you seek is the effect of a delta function.
(3) Don't throw in things that are irrelevant to your message. In this 
case, u=F(x) is irrelevant to your message.

About that limit...yes, it might be useful to have it give a DiracDelta 
result. At present what one can do, in integrating a concrete function, 
is use Limit outside Integrate to get the effect of integration against 
a delta function.

Example:

ee = Exp[-(x-xo)^2/t]/t^(1/2);

In[29]:= InputForm[Limit[Integrate[ee*x^2, {x,-Infinity,Infinity}],
          t->0, Assumptions->Element[{x,xo},Reals]]]
Out[29]//InputForm= Sqrt[Pi]*xo^2

This has the advantage, moreover, of being mathematically correct 
whereas having Limit return a DiracDelta might go against some standard 
Limit behavior (I have not fully thought this out).

Daniel Lichtblau
Wolfram Research





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