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

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=-39/8, b>=a

For fixed z and a=-39/8, b=29/8

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

Input Form

Hypergeometric2F1[-(39/8), 29/8, 5, z] == (65536 2^(1/4) (4 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (365056 - 285200 z - 554001 z^2 - 2046310 z^3 + 20734175 z^4 - 45000648 z^5 + 44425920 z^6 - 21322752 z^7 + 4055040 z^8) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (365056 - 285200 z - 554001 z^2 - 2046310 z^3 + 20734175 z^4 - 45000648 z^5 + 44425920 z^6 - 21322752 z^7 + 4055040 z^8) EllipticK[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 Sqrt[1 - z] (365056 - 285200 z - 554001 z^2 - 2046310 z^3 + 20734175 z^4 - 45000648 z^5 + 44425920 z^6 - 21322752 z^7 + 4055040 z^8) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - (730112 - 844192 z - 968967 z^2 - 3658403 z^3 - 71353205 z^4 + 308470239 z^5 - 513869664 z^6 + 433694976 z^7 - 186126336 z^8 + 32440320 z^9) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/(3751114942575 Pi (1 + Sqrt[1 - z])^(1/4) z^4)

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]