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Multiple integration: bug?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg122368] Multiple integration: bug?
  • From: "Dr. Wolfgang Hintze" <weh at snafu.de>
  • Date: Wed, 26 Oct 2011 17:41:20 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com

In calculating Integrals of the type Integrate[0<x1<x2<...<xn<1] which 
should give 1/n! I observed a strange behaviour

Using UnitStep[] it works fine for n=2 to n=5:

In[48]:=
Integrate[UnitStep[x[2] - x[1]]*UnitStep[x[3] - x[2]]*UnitStep[x[4] - 
x[3]]*UnitStep[x[5] - x[4]], {x[1], 0, 1}, {x[2], 0, 1}, {x[3], 0, 1}, 
{x[4], 0, 1}, {x[5], 0, 1}]
Out[48]=
1/120


It gives nonsense for n=6 (and higher):

In[47]:=
Integrate[UnitStep[x[2] - x[1]]*UnitStep[x[3] - x[2]]*UnitStep[x[4] - 
x[3]]*UnitStep[x[5] - x[4]]*UnitStep[x[6] - x[5]], {x[1], 0, 1}, {x[2], 
0, 1}, {x[3], 0, 1}, {x[4], 0, 1}, {x[5], 0, 1},
{x[6], 0, 1}]
Out[47]=
-(1/4)

Is this a bug or did I make a mistake somewhere (did not consider 
bounds or depth level etc.)?

Any comment is appreciated.

BTW: using Boolean instead of UnitStep seems to have no restriction.

--- Wolfgang




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