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MathGroup Archive 2007

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Re: question (for Mathematica 6!)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg75584] Re: [mg75519] question (for Mathematica 6!)
  • From: Andrzej Kozlowski <akoz at mimuw.edu.pl>
  • Date: Sat, 5 May 2007 06:13:49 -0400 (EDT)
  • References: <200705040809.EAA04730@smc.vnet.net>

On 4 May 2007, at 17:09, dimitris wrote:

> I don't have a copy of Mathematica 6 but having spent much time
> reading the online documentation (not only because of curiosity!) I
> have one question...
>
> Can somebody explain me the introduction of the HeavisideTheta
> function
>
> http://reference.wolfram.com/mathematica/ref/HeavisideTheta.html
>
> in version 6?
>
> Note that the UnitStep function is still here
>
> http://reference.wolfram.com/mathematica/ref/UnitStep.html
>
> Dimitris
>
>


I think WRI has finally decided to make a clear distinction between 
distributions (HeavisideTheta) and piecwise defined fucntions 
(UnitStep). The difference can be seen here:

In[2]:= PiecewiseExpand[HeavisideTheta[x]]
Out[2]= HeavisideTheta[x]

In[3]:= PiecewiseExpand[UnitStep[x]]
Out[3]= Piecewise[{{1, x >= 0}}]

HeavisideTheta is a distribution and not a piecewise-defined  
function so is not expanded. Another example that illustrates this:

Integrate[D[HeavisideTheta[x], x], x]
HeavisideTheta[x]
Integrate[D[UnitStep[x], x], x]
0

Note that in traditional notation both  HeavisideTheta[x] and UnitStep=

[x] look identical (at least to me!) .  It might mean that 
introduction of HeavisideTheta is a result of, on the one hand, 
recognizing that the behaviour of UnitStep was incorrect (for a 
distribution) and, on the other, of trying to preserve compatibility. =

However, since UnitStep is reamins fully documented and supported, we =

now have two superfically similar functions (the more the merrier!)  
but with a deep underlying difference in meaning.

Note also the following:

in 5.2:

Integrate[DiracDelta[x - a], {x, b, c},
   Assumptions -> a =E2=88=88 Reals && b < c]


UnitStep[a - b]*UnitStep[c - a]

In 6.0:

Integrate[DiracDelta[x - a], {x, b, c},  Assumptions -> Element[a, 
Reals] && b < c]
HeavisideTheta[a - b]*HeavisideTheta[c - a]


In TraditionalForm the outputs look identical.

Andrzej Kozlowski=


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