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Re: Re: difference between HeavisidePi and UnitBox

  • To: mathgroup at smc.vnet.net
  • Subject: [mg100425] Re: [mg100364] Re: difference between HeavisidePi and UnitBox
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Wed, 3 Jun 2009 01:11:22 -0400 (EDT)
  • References: <gvq08j$moo$1@smc.vnet.net> <gvtmcu$gc6$1@smc.vnet.net> <200906021041.GAA08533@smc.vnet.net>

It depends on your purpose. For some simple purposes they are  
equivalent but there is a huge difference when you try to do calculus  
with them. You can perform most natural operations such as  
differentiation or integration on generalized functions:

  D[HeavisidePi[x], x]
  2*DiracDelta[2*x + 1] - 2*DiracDelta[2*x - 1]

which is a well behaved generalized function while

D[UnitBox[x], x]

  Piecewise[{{Indeterminate, x == 1/2 || x == -(1/2)}}, 0]

is the zero function with two undefined values at 1/2 and -1/2,  
basically a useless object. So, for example:

Integrate[D[HeavisidePi[x], x], {x, -Infinity, 0}]
  1

while naturally

Integrate[D[UnitBox[x], x], {x, -Infinity, 0}]
0

I came up with these examples just off hand so they may look a bit  
artificial but it is easy to find ones that arise in serious problems.

Andrzej Kozlowski

On 2 Jun 2009, at 19:41, Anatoly wrote:

> I understand that it doesn't matter.  There's even some discussion
> here about it http://mathworld.wolfram.com/RectangleFunction.html
>
> My question is regarding "The piecewise version of the rectangle
> function is implemented in Mathematica as UnitBox[x], while the
> generalized function version is implemented as HeavisidePi[x]."
>
> why do we need both?  I find the integral transforms (I use mostly
> FourierTransform) actually work much better with the piecewise UnitBox
> for 5-bar patterns that I like to use.
>
> Does anyone have a thorough explanation of when generalized functions
> are better than piecewise and vice versa?
>
> -Anatoly
>
> On May 31, 6:33 pm, Jens-Peer Kuska <ku... at informatik.uni-leipzig.de>
> wrote:
>> Hi,
>>
>> what happens at x->1/2  and x->-1/2 with HeavisidePi[] and  
>> UnitBox[] ?
>>
>> For distributions like HeavisidePi[] that are used
>> inside of an integral the single
>> point does not matter. For functions like UnitBox[] it may be  
>> important.
>>
>> Regards
>>   Jens
>>
>>
>>
>> Anatoly wrote:
>>> I am a very new user, and am not quite clear on the difference  
>>> between
>>> general (HeavisidePi) and numerical (UnitBox, or piecewise)  
>>> functions.
>>
>>> When I try to create a 5-bar pattern using one of these and apply a
>>> FourierTransform, I have varying degrees of success...
>>
>>> With HeavisidePi I can create an infinite series of them and
>>> FourierTransform has no problems, but it doesn't like modifying the
>>> argument as in HeavisidePi(x/w)
>>
>>> With UnitBox I have the opposite issue.
>>
>>> Is there a clear explanation on the difference between the two  
>>> groups
>>> of functions, and when I should use which?
>
>



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