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Re: Hypergeometric Function
*To*: Joel Storch <Joel_Storch at qmlink.draper.com>
*Subject*: [mg211] Re: [mg189] Hypergeometric Function
*From*: wmm at chem.wayne.edu (Martin McClain)
*Date*: Mon, 21 Nov 94 12:01:30 EST
*Cc*: mathgroup at christensen.cybernetics.net
> Subject: Time:1:54 PM
> OFFICE MEMO Hypergeometric Function Date:11/17/94
>
>When performing the definite integral: Integrate[Sin[a x]*(Cos[x])^b,{x,0,Pi/2}] , Mathematica returned a result involving the generalized Hypergeometric function - "HyperGeometricpFq".
>When I tried to evaluate the result numerically (for specified a,b), it returned the symbolic form i.e. it did not produce a numerical value. What additional steps must be performed to enable a numerical evaluation ? (Incidently, the second edition of W
o
>lframs text does not contain any reference to the Generalized Hypergeometric Function).
Dear Joel:
Some time ago I asked this very question, and got a reply from Stephen
Wolfram himself.
Here it is:
Date: Sun, 7 Jun 1992 18:19:18 -0700
"... There is an undocumented function HypergeometricPFQ in Version 2.0 and
above which should give values for 2F2. (Get details with
?HypergeometricPFQ.)
The reason this function is undocumented, however, is that it has not
yet been tested and analysed for all choices of arguments. Jerry Keiper
(keiper at wri.com) is the person who wrote it, and he should be able to
help you make sure it gives valid results for the cases you need. ... "
Later Jerry Keiper also answered:
"Date: Tue, 9 Jun 92 10:35:11 CDT
From: keiper at wri.com
To: wmm at chem.wayne.edu
Subject: Hypergeometric2F2
Versions 2.0 and 2.1 of Mathematica provide for numerical evaluation of
general hypergeometric pFq functions. Convergence is assumed when two
successive partial sums are found to be equal so it is very possible to
trick this test and get a wrong result. It is also possible to get
completely wrong results due to very ill-conditioned sums as in
HypergeometricPFQ[{1.000001, 1.000002}, {1.000003, 1.000004}, -50] where
massive cancelation of digits occurs (the correct result would be about
8.1667992944 10^-8). The code used is:
HypergeometricPFQ[a_List, b_List, z_] :=
Module[{an = a, bn = Append[b, 1], oldsum = 0, sum = 1, term = 1},
While[oldsum != sum,
term *= Apply[Times, an++] z/Apply[Times, bn++];
oldsum = sum;
sum += term];
sum
] /; (Apply[And, Map[NumberQ, a]] && Apply[And, Map[NumberQ, b]] &&
NumberQ[z] && Precision[{a, b, z}] < Infinity &&
(Length[a] - Length[b] < 1 ||
(Length[a] - Length[b] == 1 && Abs[z] < 1)))
It would probably be considerably faster to evaluate 2F2 if you wrote your
own function hyper2F2[a,b,c,d,z] that didn't have to verify that each of
its arguments was in fact a number and that didn't try to do any symbolic
simplifications before hand. If you need to evaluate it 1000s of times
you might want to consider writing a small routine in C and connecting
to Mathematica with MathLink.
Jerry B. Keiper
keiper at wri.com"
Keiper's function, undocumented but present and waiting right there inside
in Mma, worked fine for 2F2 over a wide range, including the range we were
interested in. But we were able to push it into an area where it failed to
converge and gave wrong answers.
Regards- Martin McClain, Chemistry, Wayne State University, Deetroit
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