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MathGroup Archive 1992

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help on integration

  • To: mathgroup at yoda.physics.unc.edu
  • Subject: help on integration
  • From: thomson_p at maths.su.oz.au (Peter Thomson)
  • Date: Wed, 25 Nov 92 10:50:52 +1100

Dear Mathgroup,

I have the following function that I wish to integrate over (0,q), 

In[1]:= f[q_]=(L^a/Gamma[a])^2 Exp[-L(c s-q^2)/(c-q)](q(s-q))^(a-1)

            2 a  -1 + a         -1 + a
           L    q       (-q + s)
Out[1]= ----------------------------------
               2
         (L (-q  + c s))/(c - q)         2
        E                        Gamma[a]

In[2]:= Integrate[f[q],{q,0,s}]

Out[2]= Integrate[
 
                            2
       -(L c) - L q + (-(L c ) + L c s)/(-c + q)  2 a  -1 + a         -1 + a
      E                                          L    q       (-q + s)
>     ----------------------------------------------------------------------, 
                                            2
                                    Gamma[a]
 
>    {q, 0, s}]

Mma not surprisingly cannot perform this integration directly.
I was wondering if anyone can recognise this as being of a standard 
form that I can coerce Mma to perform.  Any tips would be most
appreciated.

Incidentally, The application arose from the convolution of two
gamma random variables (correlated, not independent).  I noticed
that the ContinuousDistributions.m Package uses an unusual 
definition of the parameter lambda which needs to be inverted to
obtain the more usual definition for the density, i.e.

In[3]:= <<DataAnalysis/ContinuousDistributions.m

In[4]:= Density[GammaDistribution[alpha,1/lambda],t]

              alpha  -1 + alpha
        lambda      t
Out[4]= -----------------------
         lambda t
        E         Gamma[alpha]


Peter Thomson,

School of Mathematics and Statistics,
University of Sydney  NSW  2006
Australia

email: thomson_p at maths.su.oz.au





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