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MathGroup Archive 1995

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

  • To: mathgroup at christensen.cybernetics.net
  • Subject: [mg502] Re: [mg497] Numerical evaluation of HypergeometricPFQ[{a,b,c},{d,e},1]
  • From: bob Hanlon <hanlon at pafosu2.hq.af.mil>
  • Date: Fri, 3 Mar 1995 06:10:55

Take a look at "Higher Transcendental Functions, Volume I", Erdelyi, A. et
al., McGraw-Hill, 1953; section 4.4, Generalized hypergeometric series with
unit argument in the case p = q + 1, pp. 188-190.  It provides a few results.

Bob Hanlon
hanlon at pafosu2.hq.af.mil

On Wed, 1 Mar 1995, NELSON M. BLACHMAN wrote:

>   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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