Numerical Evaluation of HypergeometricPFQ

*To*: mathgroup at christensen.cybernetics.net*Subject*: [mg492] Numerical Evaluation of HypergeometricPFQ*From*: "NELSON M. BLACHMAN" <blachman at gtewd.mtv.gtegsc.com>*Date*: Sun, 26 Feb 1995 23:54:13 PST

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 HypergeomPFQ. When I asked it to plot the result, however, I found it apparently unable to determine numerical values for HypergeomPFQ. Maple can sometimes compute numerical values for HypergeomPFQ, 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 HypergeomPFQ numerically? I suspect there's a simple expression for HypergeomPFQ[{ },{ },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.]