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=

**References**:**question (for Mathematica 6!)***From:*dimitris <dimmechan@yahoo.com>