       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:=
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:=
p5 = (4!*Integrate[Integrate[Integrate[Integrate[UnitStep, {\[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= 1

How would you proceed to solve In? 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

```

• Prev by Date: Simplifying results
• Next by Date: Re: New MathGL3d Add-On for Mathematica Available
• Previous by thread: Simplifying results
• Next by thread: Re: Multiple integration of UnitStep fails