Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b8qd)

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=-41/8, b>=a

For fixed z and a=-41/8, b=-29/8

http://functions.wolfram.com/07.23.03.b8qd.01

Input Form

Hypergeometric2F1[-(41/8), -(29/8), 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-36428677120 + 792466026880 z - 9161886780585 z^2 + 82047351846075 z^3 - 822867068648625 z^4 - 50346603239453721 z^5 - 133049683844883711 z^6 - 89881197245254863 z^7 - 16193203793756955 z^8 - 436908761491475 z^9 + 4211168785460 z^10) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (-36428677120 + 806126780800 z - 9455326178265 z^2 + 85403785076610 z^3 - 852737244270450 z^4 - 12470688612027846 z^5 - 10072506065091300 z^6 + 13126449708963558 z^7 + 9217653775261218 z^8 + 974057487966430 z^9 + 1052792196365 z^10) EllipticK[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (-36428677120 + 792466026880 z - 9161886780585 z^2 + 82047351846075 z^3 - 822867068648625 z^4 - 50346603239453721 z^5 - 133049683844883711 z^6 - 89881197245254863 z^7 - 16193203793756955 z^8 - 436908761491475 z^9 + 4211168785460 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-36428677120 + 792466026880 z - 9161886780585 z^2 + 82047351846075 z^3 - 822867068648625 z^4 - 50346603239453721 z^5 - 133049683844883711 z^6 - 89881197245254863 z^7 - 16193203793756955 z^8 - 436908761491475 z^9 + 4211168785460 z^10) EllipticK[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (9850134399725275130475 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) z^5)

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]