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MathGroup Archive 2005

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Multiple integration of UnitStep fails

  • To: mathgroup at smc.vnet.net
  • Subject: [mg63194] Multiple integration of UnitStep fails
  • From: "Dr. Wolfgang Hintze" <weh at snafu.de>
  • Date: Sat, 17 Dec 2005 03:46:34 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Hello group,

trying to solve the nice problem of determining the probability pn that 
a polygon formed by n (>=4) random points on the unit circle is void of 
an acute angle I came up with the following multiple integral (written 
down here for n=5)

In[15]:=
p5 = (4!*Integrate[Integrate[Integrate[Integrate[UnitStep[\[Phi]4 - Pi, 
\[Phi]5 - \[Phi]2 - Pi, Pi - \[Phi]3, Pi - \[Phi]4 + \[Phi]2,
          Pi - \[Phi]5 + \[Phi]3], {\[Phi]5, \[Phi]4, 2*Pi}], {\[Phi]4, 
\[Phi]3, 2*Pi}], {\[Phi]3, \[Phi]2, 2*Pi}], {\[Phi]2, 0, 2*Pi}])/(2*Pi)^4

Mathematica version 4 was not able to solve this but returned it 
unevaluated after some minutes; version 5 complained several things 
like: argument is not a power series, unable to check convergence, but 
didn't come up with any result in ten minutes (I wouldn't wait longer).

I could successfully check the normalization at least:

In[14]:=
p5 = (4!*Integrate[Integrate[Integrate[Integrate[UnitStep[1], {\[Phi]5, 
\[Phi]4, 2*Pi}], {\[Phi]4, \[Phi]3, 2*Pi}], {\[Phi]3, \[Phi]2, 2*Pi}],
      {\[Phi]2, 0, 2*Pi}])/(2*Pi)^4

Out[14]= 1

How would you proceed to solve In[15]? What about the general case (n=6 
see below)?

Any hints are greatly appreciated.

Regards,
Wolfgang

PS:

Here's the probability for the case n=6

p6 = 
(5!*Integrate[Integrate[Integrate[Integrate[Integrate[UnitStep[\[Phi]5 - 
Pi, \[Phi]6 - \[Phi]2 - Pi, Pi - \[Phi]3, Pi - \[Phi]4 + \[Phi]2,
           Pi - \[Phi]5 + \[Phi]3, Pi - \[Phi]6 + \[Phi]4], {\[Phi]6, 
\[Phi]5, 2*Pi}], {\[Phi]5, \[Phi]4, 2*Pi}], {\[Phi]4, \[Phi]3, 2*Pi}],
       {\[Phi]3, \[Phi]2, 2*Pi}], {\[Phi]2, 0, 2*Pi}])/(2*Pi)^5

$Aborted


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