Re: Beta function, Integral

*To*: mathgroup at smc.vnet.net*Subject*: [mg79945] Re: Beta function, Integral*From*: Asim <maa48 at columbia.edu>*Date*: Thu, 9 Aug 2007 05:23:08 -0400 (EDT)*References*: <f990ds$btq$1@smc.vnet.net><f9c1q6$75a$1@smc.vnet.net>

On Aug 8, 2:19 pm, dimitris <dimmec... at yahoo.com> wrote: > 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[c > > Out[47]= > c > > Regards > Dimitris Thanks for all who responded to by Beta function integral post. I used Mathematica 6.0, and by mistake used t^{p-1} in the integral, instead of t^(p-1). I was expecting (Gamma[p] Gamma[q])/Gamma[p + q] which can be obtained from Mathematica 5.2 as the answer, but Mathematica 6.0 returns (\[Pi] Csc[\[Pi] q] Gamma[p])/(Gamma[1 - q] Gamma[p + q]). The two are indeed the same, but the former is the more "standard" answer given the definition of the Beta function. Regards Asim Ansari