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MathGroup Archive 2000

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Integrating Unit Step Functions (Convolution)

  • To: mathgroup at smc.vnet.net
  • Subject: [mg22926] Integrating Unit Step Functions (Convolution)
  • From: Roger Jones <rmj at slac.stanford.edu>
  • Date: Thu, 6 Apr 2000 02:04:43 -0400 (EDT)
  • Organization: Stanford Linear Accelerator Center
  • Sender: owner-wri-mathgroup at wolfram.com

In mathematica 3.1 the convolution of II[x], (the "Top Hat" function)
with itself gives
(the Triangle function):

In[]=Simplify[Integrate[ II[u] II[x-u],{u,-Infinity,Infinity}]=
Out[]=(-1 + x) UnitStep[-1 + x] - 2 x UnitStep[x] +  (1 + x) UnitStep[1
+ x]


whereas Mathematica 4.0 gives:

Out[]=
   1
If[- + x < 0, (-1 + x) UnitStep[-1 + x] -
   2

   2 x UnitStep[x] + (1 + x) UnitStep[1 + x],

                         1
  Integrate[(-UnitStep[-(-) + u] +
                         2

               1
      UnitStep[- + u])
               2

                 1                      1
    (-UnitStep[-(-) - u + x] + UnitStep[- - u + x])
                 2                      2

    , {u, -Infinity, Infinity}]]


Is there some bug in Mathematica 4.0 or perhaps there is some options I
should apply for this integral?
By hand, Mathematica 3.1 gives the correct result.

-Roger Jones



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