Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b8q5)

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.b8q5.01

Input Form

Hypergeometric2F1[-(41/8), -(29/8), 4, z] == (2048 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-142299520 + 3818741025 z - 76452640550 z^2 - 8090505356845 z^3 - 31960734316680 z^4 - 30126517177001 z^5 - 7217191753870 z^6 - 249878689515 z^7 + 3001545820 z^8) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (-142299520 + 3872103345 z - 77870077175 z^2 - 2264640719395 z^3 - 3894453370635 z^4 + 2411971094179 z^5 + 3333411392171 z^6 + 487101490575 z^7 + 750386455 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (-142299520 + 3818741025 z - 76452640550 z^2 - 8090505356845 z^3 - 31960734316680 z^4 - 30126517177001 z^5 - 7217191753870 z^6 - 249878689515 z^7 + 3001545820 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-142299520 + 3818741025 z - 76452640550 z^2 - 8090505356845 z^3 - 31960734316680 z^4 - 30126517177001 z^5 - 7217191753870 z^6 - 249878689515 z^7 + 3001545820 z^8) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (5936720604940125 Pi (1 + Sqrt[1 - z])^(1/4) (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]