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MathGroup Archive 2007

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Re: Beta function, Integral

  • To: mathgroup at smc.vnet.net
  • Subject: [mg80002] Re: Beta function, Integral
  • From: Jean-Marc Gulliet <jeanmarc.gulliet at gmail.com>
  • Date: Fri, 10 Aug 2007 01:54:53 -0400 (EDT)
  • Organization: The Open University, Milton Keynes, UK
  • References: <f990ds$btq$1@smc.vnet.net><f9c0i1$5rq$1@smc.vnet.net> <f9eljl$j7e$1@smc.vnet.net>

dimitris wrote:
> 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

Hi Dimitris,

You are right, of course. I did not spot at all the erroneous curly 
braces in the original expression, though I spent some time typing in 
the result by hand in v6 an v5.2! Some vacations are needed here :-)

Best regards,
-- 
Jean-Marc


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