Re: Numerical evaluation of HypergeometricPFQ[{a,b,c},{d,e},1]
- To: mathgroup at christensen.cybernetics.net
- Subject: [mg508] Re: [mg497] Numerical evaluation of HypergeometricPFQ[{a,b,c},{d,e},1]
- From: bob Hanlon <hanlon at pafosu2.hq.af.mil>
- Date: Sun, 5 Mar 1995 14:27:27
The specific result (Integrate[Sin[u t]/Sin[t], {t, 0, Pi/2}]) is given below.
Bob Hanlon
hanlon at pafosu2.hq.af.mil
____________________
Integrate[Sin[u t]/Sin[t], {t, 0, Pi/2}] // Simplify
Pi u 1 3 - u 3 + u
u Cos[----] HypergeometricPFQ[{-, 1, 1}, {-----, -----}, 1]
2 2 2 2
-----------------------------------------------------------
2
1 - u
Clear[iss];
iss::usage = "iss[u] evaluates the integral: \n
Integrate[Sin[u t]/Sin[t], {t, 0, Pi/2}]";
iss[ u_ /; u == 0 ] = 0;
iss[ u_ /; IntegerQ[(u - 1)/2] ] := Pi/2 Sign[u];
(* general expression is indeterminate for odd integers *)
(* Gradshteyn & Ryzhik, 3.612.3 *)
iss[ u_ /; IntegerQ[u/2] ] := 2 Sum[(-1)^(k-1) / (2k - 1),
{k, u/2 Sign[u]}]; (* Gradshteyn & Ryzhik, 3.612.4 *)
iss[ u_ ] := - Cos[u Pi/2] (PolyGamma[(1 - u)/4] -
PolyGamma[(3 - u)/4] - PolyGamma[(1 + u)/4] +
PolyGamma[(3 + u)/4]) / 4;
(* PolyGamma[z] = PolyGamma[0, z] is the digamma function,
that is, the logarithmic derivative of the gamma function *)
Plot[{iss[u], Pi/2 Sign[u]}, {u, -5, 11}];
_______________
> 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.]
> >
> >
>
>