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????

(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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