Watson's Theorem

Author

Eric W. Weisstein
August 17, 2005

This notebook downloaded from http://mathworld.wolfram.com/notebooks/SpecialFunctions/WatsonsTheorem.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/WatsonsTheorem.html.

©2005 Wolfram Research, Inc. except for portions noted otherwise

Identity

HypergeometricPFQ[{a,b,c},{(1/2)(a+b+1),2c},1]//FunctionExpand

For convergence, requires Re((Subscript[b, 1])+(Subscript[b, 2])+1-(Subscript[a, 1])-(Subscript[a, 2])-(Subscript[a, 3]))>1, which implies Re (-(a/2)-(b/2)+c+(3/2))>1

(values=Select[Table[Random[Complex,5(I+1){-1,1},30],{100},{3}],Re[-#[[1]]/2-#[[2]]/2+#[[3]]+3/2]>1&])//Length//Timing

Subtract@@@({HypergeometricPFQ[{a,b,c},{(1/2)(a+b+1),2c},1],
(Gamma[(1/2)]Gamma[(1/2)+c]Gamma[(1/2)(1+a+b)]Gamma[(1/2)-(1/2)a-(1/2)b+c])/(Gamma[(1/2)(1+a)]Gamma[(1/2)(1+b)]Gamma[(1/2)-(1/2)a+c]Gamma[(1/2)-(1/2)b+c])}/.Thread[{a,b,c}->#]&/@values)//Chop//Union//Timing