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From: ran@ho95b.UUCP (RANeinast)
Newsgroups: net.math
Subject: Re: Integration Problem
Message-ID: <308@ho95b.UUCP>
Date: Mon, 11-Feb-85 15:38:50 EST
Article-I.D.: ho95b.308
Posted: Mon Feb 11 15:38:50 1985
Date-Received: Tue, 12-Feb-85 06:38:50 EST
Organization: AT&T-Bell Labs, Holmdel, NJ
Lines: 58

>Given n variables  X  such that :
>                    i
>
>
>           n
>       \--------
>        \            X     =  1 
>        /             i
>       /--------
>         i = 1
>
>
>   and n constants   S    such that :
>                      i
>
>           n
>       \--------
>        \            S     =  m       for some m > 0
>        /             i
>       /--------
>         i = 1
>
>WHAT IS :
>
>
>         1        1
>        S        S
>        S        S     S      S      S          S
>        S        S      1      2      3          n     
>        S ....   S    X      X      X  ....    X     dX  dX  dX  ... dX 
>        S        S     1      2      3          n      1   2   3       n
>        S        S
>       0        0


This is a generalization of the Beta function.  Using G(x)
for the Gamma function [BTW, G(x)=(x-1)!], and letting Si=Ti-1:


                    G(T1)*G(T2)* ... *G(Tn)
         Integral=  -----------------------
                       G(T1+T2+ ... +Tn)      .


The above is actually fairly easy to show (but takes up lots of space).
After the constraint on the X's is inserted, one must be careful
of what happens to the limits of integration, but a simple recursion
relation can be derived.  Note that the answer must be symmetric
with respect to the Ti's.

If anybody is *really* interested, I can post or mail a more complete
derivation.

-- 

". . . and shun the frumious Bandersnatch."
Robert Neinast (ihnp4!ho95c!ran)
AT&T-Bell Labs