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MathGroup Archive 2007

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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[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])}
>
> 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[18]:=
$Version

Out[18]=
"5.2 for Microsoft Windows (June 20, 2005)"

In[20]:=
Integrate[t^(p - 1)*(1 - t)^(q - 1), {t, 0, 1}, Assumptions -> {p > 0,
q > 0}]
FunctionExpand[Beta[p, q] - %]

Out[20]=
(Gamma[p]*Gamma[q])/Gamma[p + q]

Out[21]=
0

Ommiting { } from your output, we have also

In[28]:=
FullSimplify[Beta[p, q] - (Pi*Csc[Pi*q]*Gamma[p])/(Gamma[1 -
q]*Gamma[p + q])]

Out[28]=
0

as it must be.

Note also that

In[30]:= Beta[p, q] // FunctionExpand
Out[30]= (Gamma[p] Gamma[q])/Gamma[p + q]

and

In[46]:=
FullSimplify[(Gamma[p]*Gamma[q])/Gamma[p + q] == Beta[p, q]]

Out[46]=
True

but

In[47]:=
FullSimplify[(Gamma[p]*Gamma[q])/Gamma[p + q]]

Out[47]=
(Gamma[p]*Gamma[q])/Gamma[p + q]

Regards
Dimitris



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