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Re: Multiple integration: bug?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg122399] Re: Multiple integration: bug?
  • From: Ray Koopman <koopman at sfu.ca>
  • Date: Thu, 27 Oct 2011 06:32:23 -0400 (EDT)
  • Delivered-to: l-mathgroup@mail-archive0.wolfram.com
  • References: <j89vct$224$1@smc.vnet.net>

On Oct 26, 2:51 pm, "Dr. Wolfgang Hintze" <w... at snafu.de> wrote:
> 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

Why not just integrate over 0 < x1 < x2 < ... < xn < 1 ?

In[1]:=
f[n_Integer?Positive] := Block[{x}, Integrate[1, Sequence @@
({#2,#1,1}& @@@ Partition[Prepend[Array[x,n],0],2,1]) ] ]

In[2]:= f[6]

Out[2]= 1/720



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