Re: Beta function, Integral

*To*: mathgroup at smc.vnet.net*Subject*: [mg79917] Re: Beta function, Integral*From*: dimitris <dimmechan at yahoo.com>*Date*: Thu, 9 Aug 2007 05:08:32 -0400 (EDT)*References*: <f990ds$btq$1@smc.vnet.net><f9c0i1$5rq$1@smc.vnet.net>

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