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Re: Beta function, Integral
*To*: mathgroup at smc.vnet.net
*Subject*: [mg79893] Re: Beta function, Integral
*From*: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
*Date*: Wed, 8 Aug 2007 04:50:02 -0400 (EDT)
*Organization*: The Open University, Milton Keynes, UK
*References*: <f990ds$btq$1@smc.vnet.net>
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
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