       Re: Beta function, Integral

• To: mathgroup at smc.vnet.net
• Subject: [mg79914] Re: Beta function, Integral
• From: dimitris <dimmechan at yahoo.com>
• Date: Wed, 8 Aug 2007 05:00:54 -0400 (EDT)
• References: <f990ds\$btq\$1@smc.vnet.net>

```On 7    , 08:37, Asim <ma... at columbia.edu> wrote:
> Hi
>
> 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:= Integrate[t^{p - 1}*(1 - t)^(q - 1), {t, 0, 1},  Assumptions -
>
> > {p > 0, q > 0}]
>
> Out= {(\[Pi] Csc[\[Pi] q] Gamma[p])/(Gamma[1 - q] Gamma[p + q])}
>
> Thanks
>
> Asim Ansari

First note that you used List where you should have used parentheses!
A common mistake.
It must be t^(p-1); not {p-1}.

In the Mathematica I work, I took:

In:=
\$Version

Out=
"5.2 for Microsoft Windows (June 20, 2005)"

In:=
Integrate[t^(p - 1)*(1 - t)^(q - 1), {t, 0, 1}, Assumptions -> {p > 0,
q > 0}]
FunctionExpand[Beta[p, q] - %]

Out=
(Gamma[p]*Gamma[q])/Gamma[p + q]

Out=
0

Ommiting { } from your output, we have also

In:=
FullSimplify[Beta[p, q] - (Pi*Csc[Pi*q]*Gamma[p])/(Gamma[1 -
q]*Gamma[p + q])]

Out=
0

as it must be.

Note also that

In:= Beta[p, q] // FunctionExpand
Out= (Gamma[p] Gamma[q])/Gamma[p + q]

and

In:=
FullSimplify[(Gamma[p]*Gamma[q])/Gamma[p + q] == Beta[p, q]]

Out=
True

but

In:=
FullSimplify[(Gamma[p]*Gamma[q])/Gamma[p + q]]

Out=
(Gamma[p]*Gamma[q])/Gamma[p + q]

Regards
Dimitris

```

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