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Re: Another Mathematica 6 bug?

  • To: mathgroup at smc.vnet.net
  • Subject: [mg86360] Re: [mg86284] Another Mathematica 6 bug?
  • From: Bob Hanlon <hanlonr at cox.net>
  • Date: Sun, 9 Mar 2008 05:03:55 -0500 (EST)
  • Reply-to: hanlonr at cox.net

A temporary change of variables works well

Integrate[p*p^X*(1 - p)^(n - X)*p^(X - 1/2)*(1 - p)^(n - X - 1/2) /. 
   X -> (x - 1/2)/2, {p, 0, 1}, GenerateConditions -> False] /. x -> 2 X + 1/2

(Gamma[2*n - 2*X + 1/2]*
      Gamma[2*X + 3/2])/Gamma[2*n + 2]


Bob Hanlon

---- Bob Hanlon <hanlonr at cox.net> wrote: 
> They appear to be equivalent for Re[n-X] > -1/4
> 
> sol = FullSimplify[
>   Integrate[p*p^X*(1 - p)^(n - X)*p^(X - 1/2)*(1 - p)^(n - X - 1/2), {p, 0, 
>     1}, GenerateConditions -> False]]
> 
> (1/2)*Pi*(4*X + 1)*
>    Hypergeometric2F1[-2*n, 
>      2*X + 3/2, 2, 1]*Sec[2*Pi*X]
> 
> See http://functions.wolfram.com/07.23.03.0002.01
> 
> For the stated condition
> 
> sol2 = FullSimplify[
>   sol /. Hypergeometric2F1[a_, b_, c_, 1] ->
>     
>     Gamma[c] Gamma[c - a - b]/(Gamma[c - a] Gamma[c - b])]
> 
> -((Pi*Gamma[2*n - 2*X + 1/2]*
>          Sec[2*Pi*X])/(Gamma[2*n + 2]*
>          Gamma[-2*X - 1/2]))
> 
> Which is equivalent to the result that you were expecting
> 
> sol2 == Gamma[1/2 + 2*n - 2*X]*Gamma[3/2 + 2*X]/Gamma[2 + 2*n] // FullSimplify
> 
> True
> 
> 
> Bob Hanlon
> 
> ---- Francogrex <franco at grex.org> wrote: 
> > This integration below:
> > FullSimplify[Integrate[p*p^X*(1 - p)^(n - X)*p^(X - 1/2)*
> >     (1 - p)^(n - X - 1/2), {p, 0, 1}, GenerateConditions -> False]]
> > 
> > Should give:
> > (Gamma[1/2 + 2*n - 2*X]*Gamma[3/2 + 2*X])/Gamma[2 + 2*n]
> > 
> > Instead in mathematica 6, it's giving:
> > (1/2)*Pi*(4*X + 1)*Hypergeometric2F1[-2*n, 2*X + 3/2, 2,
> > 1]*Sec[2*Pi*X]
> > 
> > Something wrong?
> > 



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