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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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