Re: Beta function, Integral

*To*: mathgroup at smc.vnet.net*Subject*: [mg79914] Re: Beta function, Integral*From*: dimitris <dimmechan at yahoo.com>*Date*: Wed, 8 Aug 2007 05:00:54 -0400 (EDT)*References*: <f990ds$btq$1@smc.vnet.net>

On 7 , 08:37, Asim <ma... at columbia.edu> wrote: > Hi > > 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])} > > Thanks > > Asim Ansari First note that you used List where you should have used parentheses! A common mistake. It must be t^(p-1); not {p-1}. In the Mathematica I work, I took: In[18]:= $Version Out[18]= "5.2 for Microsoft Windows (June 20, 2005)" In[20]:= Integrate[t^(p - 1)*(1 - t)^(q - 1), {t, 0, 1}, Assumptions -> {p > 0, q > 0}] FunctionExpand[Beta[p, q] - %] Out[20]= (Gamma[p]*Gamma[q])/Gamma[p + q] Out[21]= 0 Ommiting { } from your output, we have also In[28]:= FullSimplify[Beta[p, q] - (Pi*Csc[Pi*q]*Gamma[p])/(Gamma[1 - q]*Gamma[p + q])] Out[28]= 0 as it must be. Note also that In[30]:= Beta[p, q] // FunctionExpand Out[30]= (Gamma[p] Gamma[q])/Gamma[p + q] and In[46]:= FullSimplify[(Gamma[p]*Gamma[q])/Gamma[p + q] == Beta[p, q]] Out[46]= True but In[47]:= FullSimplify[(Gamma[p]*Gamma[q])/Gamma[p + q]] Out[47]= (Gamma[p]*Gamma[q])/Gamma[p + q] Regards Dimitris