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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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