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