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