Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b8ql)

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=-27/8

http://functions.wolfram.com/07.23.03.b8ql.01

Input Form

Hypergeometric2F1[-(41/8), -(27/8), 4, z] == (1/(94266801003375 Pi z^3)) (1024 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (-4 (-2597760 + 67156155 z - 1285160580 z^2 - 142690414255 z^3 - 532150133610 z^4 - 458176756419 z^5 - 93110780928 z^6 - 1696844457 z^7 + 30573774 z^8) EllipticE[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (-324720 + 8341245 z - 18570777390 z^2 - 87408533405 z^3 - 101752283700 z^4 - 35120232333 z^5 - 2964276942 z^6 + 5095629 z^7) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 6 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (162360 - 4079295 z - 15265533690 z^2 - 64324242485 z^3 - 60426123300 z^4 - 13330190841 z^5 - 281958138 z^6 + 5095629 z^7) EllipticK[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 2 (-2597760 + 67156155 z - 1285160580 z^2 - 142690414255 z^3 - 532150133610 z^4 - 458176756419 z^5 - 93110780928 z^6 - 1696844457 z^7 + 30573774 z^8) EllipticK[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))

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]