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http://functions.wolfram.com/07.23.17.0108.01
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Hypergeometric2F1[a, b, 1/2, z] ==
(((Gamma[a + 1/2] Gamma[1 - b])/(2 Sqrt[Pi] Gamma[a - b + 1]))
(Hypergeometric2F1[2 a, 1 - 2 b, a - b + 1, (Sqrt[1 - z] + Sqrt[-z])/
(2 Sqrt[1 - z])] + Hypergeometric2F1[2 a, 1 - 2 b, a - b + 1,
(Sqrt[1 - z] - Sqrt[-z])/(2 Sqrt[1 - z])]))/(1 - z)^a /;
!IntervalMemberQ[Interval[{1, Infinity}], z]
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HypergeometricPFQ[{},{},z] | HypergeometricPFQ[{},{b},z] | HypergeometricPFQ[{a},{},z] | HypergeometricPFQ[{a},{b},z] | HypergeometricPFQ[{a1},{b1,b2},z] | HypergeometricPFQ[{a1,a2},{b1,b2},z] | HypergeometricPFQ[{a1,a2},{b1,b2,b3},z] | HypergeometricPFQ[{a1,a2,a3},{b1,b2},z] | HypergeometricPFQ[{a1,a2,a3,a4},{b1,b2,b3},z] | HypergeometricPFQ[{a1,a2,a3,a4,a5},{b1,b2,b3,b4},z] | HypergeometricPFQ[{a1,a2,a3,a4,a5,a6},{b1,b2,b3,b4,b5},z] | HypergeometricPFQ[{a1,...,ap},{b1,...,bq},z] | |
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