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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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