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MathGroup Archive 1995

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Numerical evaluation of HypergeometricPFQ[{a,b,c},{d,e},1]

  • To: mathgroup at christensen.cybernetics.net
  • Subject: [mg497] Numerical evaluation of HypergeometricPFQ[{a,b,c},{d,e},1]
  • From: "NELSON M. BLACHMAN" <blachman at gtewd.mtv.gtegsc.com>
  • Date: Wed, 01 Mar 1995 22:28:48 PST

  This is a postscript to my 26 February message (reproduced below) 
complaining about Mma's not giving a numerical value for 
HypergeometricPFQ[{1/2,1,1},{1.45,1.55},1].  

  I've since found that Mma does evaluate things like 
HypergeometricPFQ[{1/2,1,1},{1.4,1.6},0.99999],  but it takes 
something like an hour on my 486DX33 PC--and it takes longer 
and longer as the last argument gets closer and closer to 1.  
So it's good that Mma quickly announces its inability to 
compute  HypergeometricPFQ[{1/2,1,1},{1.4,1.6},1].  

  A different method is evidently needed when the last argument is 
1--provided that the sum of the components of the middle argument 
exceeds the sum of the components of the first argument.  (If not, 
the hypergeometric function seems likely to be infinite.)  So I 
continue to hope for a formula for  3F2[{a,b,c},{d,e},1]  as a 
ratio of products of gamma functions of (e, f, and e + f minus 
0, a, b, c, a + b, etc.), though I've so far been unable to 
devise a satisfactory conjecture of this sort.

						Nelson M. Blachman


>From:	GTEWD::BLACHMAN     "NELSON M. BLACHMAN" 26-FEB-1995 23:54:14.57
>To:	MX%"mathgroup at christensen.cybernetics.net"
>CC:	BLACHMAN
>Subj:	Numerical Evaluation of HypergeometricPFQ

  I was pleased to see just now that Mma's able to evaluate 
Integrate[Sin[t u]/Sin[t],{t,0,Pi/2}]  in terms of HypergeomtricPFQ.  
When I asked it to plot the result, however, I found it apparently 
unable to determine numerical values for HypergeomtricPFQ.  

  Maple can sometimes compute numerical values for HypergeomtricPFQ, 
but for the particular denominator indices here it complains of 
iteration limits' being exceeded.  Does anyone know of a way to 
get Mma 2.2 to evaluate HypergeomtricPFQ numerically?

  I suspect there's a simple expression for HypergeomtricPFQ[{ },{ },1] 
in terms of gamma functions when  P = Q + 1;  there is, anyhow, if  
P = 2  and  Q = 1.  Does anyone know if that's true and, if so, what 
it is?
					Nelson M. Blachman
					GTE Government Systems Corp.
					Mountain View, California

Mathematica 2.2 for DOS 387
Copyright 1988-93 Wolfram Research, Inc.

In[1]:= Integrate[Sin[t u]/Sin[t],{t,0,Pi/2}]

         3   u        Pi u                     1          3 - u  3 + u
        (- + -) u Cos[----] HypergeometricPFQ[{-, 1, 1}, {-----, -----}, 1]
         2   2         2                       2            2      2
Out[1]= -------------------------------------------------------------------
                               3    u     1    u   1   u
                          4 (-(-) - -) (-(-) - -) (- - -)
                               2    2     2    2   2   2

In[2]:= f[t_]:= %1 /. u -> t     Simplify ELIMINATES THE FIRST FACTOR IN 
				 THE NUMERATOR AND IN THE DENOMINATOR.  
In[3]:= f[.1] // N

Out[3]= 0.0997665 HypergeometricPFQ[{0.5, 1., 1.}, {1.45, 1.55}, 1.]



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