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MathGroup Archive 2007

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Re: Beta function, Integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg79917] Re: Beta function, Integral
  • From: dimitris <dimmechan at yahoo.com>
  • Date: Thu, 9 Aug 2007 05:08:32 -0400 (EDT)
  • References: <f990ds$btq$1@smc.vnet.net><f9c0i1$5rq$1@smc.vnet.net>

On 8    , 11:57, Jean-Marc Gulliet <jeanmarc.gull... at gmail.com> wrote:
> Asim wrote:
> > The following integral does not seem to give the correct answer. The
> > answer should be the Euler Beta function,  Beta[p,q]. Can anybody let
> > me know what I am doing wrong? Or is this a bug?
>
> > In[12]:= Integrate[t^{p - 1}*(1 - t)^(q - 1), {t, 0, 1},  Assumptions -
> >> {p > 0, q > 0}]
>
> > Out[12]= {(\[Pi] Csc[\[Pi] q] Gamma[p])/(Gamma[1 - q] Gamma[p + q])}
>
> The /mathematical/ expression that is returned by Mathematica 6 is
> correct, indeed. However, the answer has funny extra curly brackets and
> if one try to use the expression to check its validity against Beta[p,q]
> the expression returned unevaluated. On the other hand, if one type in
> the expression by hand, the simplification occurs and the identity is
> checked positively.
>
> In[1]:= $Version
>
> Out[1]= "6.0 for Microsoft Windows (32-bit) (June 19, 2007)"
>
> In[2]:= sol =
>   Integrate[t^{p - 1}*(1 - t)^(q - 1), {t, 0, 1},
>    Assumptions -> {p > 0, q > 0}]
>
> Out[2]= {(\[Pi] Csc[\[Pi] q] Gamma[p])/(Gamma[1 - q] Gamma[p + q])}
> --------^---------------------------------------------------------^
> Note the spurious curly brackets.
>
> In[3]:= FullSinplify[Beta[p, q] == sol[[1]]]
>
> Out[3]= FullSinplify[
>   Beta[p, q] == (\[Pi] Csc[\[Pi] q] Gamma[p])/(
>    Gamma[1 - q] Gamma[p + q])]
>
> Even though we took out the contains of the list, Mathematica returns
> the FullSimplify unevaluated.
>
> In[4]:= FullSimplify[
>   Beta[p, q] == (Pi Csc[Pi q] Gamma[p])/(Gamma[1 - q] Gamma[p + q])]
>
> Out[4]= True
>
> Now, having entered the expression by hand, Mathematica is able to check
> the identity.
>
> A similar behavior can be seen with Mathematica 5.2, though the
> expression returned is different.
>
> In[1]:=
> $Version
>
> Out[1]=
> "5.2 for Microsoft Windows (June 20, 2005)"
>
> In[2]:=
> sol = Integrate[t^{p - 1}*(1 - t)^(q - 1), {t, 0, 1},
>     Assumptions -> {p > 0, q > 0}]
>
> Out[2]=
> {(Gamma[p]*Gamma[q])/Gamma[p + q]}
>
> In[3]:=
> FullSinplify[Beta[p, q] == sol[[1]]]
>
> Out[3]=
> FullSinplify[Beta[p, q] == (Gamma[p]*Gamma[q])/
>      Gamma[p + q]]
>
> In[4]:=
> FullSimplify[Beta[p, q] == (Gamma[p]*Gamma[q])/
>      Gamma[p + q]]
>
> Out[4]=
> True
>
> --
> Jean-Marc

Hi Jean Marc.

I think the person who made the question should
wanted to write

 Integrate[t^(p - 1)*(1 - t)^(q - 1), {t, 0, 1},Assumptions -> {p > 0,
q > 0}]

instead of

 Integrate[t^{p - 1}*(1 - t)^(q - 1), {t, 0, 1},Assumptions -> {p > 0,
q > 0}]

So that's explain the presence of the "spurious" curl brackets.

Regards
Dimitris



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