Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b5ru)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 8 and fixed z and a<0

For fixed z and a=-47/8, b>=a

For fixed z and a=-47/8, b=45/8

http://functions.wolfram.com/07.23.03.b5ru.01

Input Form

Hypergeometric2F1[-(47/8), 45/8, 4, z] == (2048 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (55762304 + 667840719 z + 9496146114 z^2 - 381274908665 z^3 + 2101934615088 z^4 - 4830901979904 z^5 + 5533044228096 z^6 - 3137180073984 z^7 + 704136609792 z^8) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (55762304 + 667840719 z + 9496146114 z^2 - 381274908665 z^3 + 2101934615088 z^4 - 4830901979904 z^5 + 5533044228096 z^6 - 3137180073984 z^7 + 704136609792 z^8) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + Sqrt[1 - z] (55762304 + 667840719 z + 9496146114 z^2 - 381274908665 z^3 + 2101934615088 z^4 - 4830901979904 z^5 + 5533044228096 z^6 - 3137180073984 z^7 + 704136609792 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (55762304 + 646929855 z + 9239988030 z^2 + 401214849931 z^3 - 4312236794712 z^4 + 15879476023488 z^5 - 28742187048960 z^6 + 27845602836480 z^7 - 13898315464704 z^8 + 2816546439168 z^9) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (804989266676595 Pi (1 + Sqrt[1 - z])^(1/4) z^3)

Standard Form

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MathML Form

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Rule Form

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Date Added to functions.wolfram.com (modification date)

2007-05-02

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,...,a_p},{b₁,...,b_q},z]