Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b9fn)

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=35/8

http://functions.wolfram.com/07.23.03.b9fn.01

Input Form

Hypergeometric2F1[-(41/8), 35/8, 5, z] == (65536 2^(1/4) (-4 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (1784320 + 2174640 z + 5907075 z^2 + 32619600 z^3 - 2009671209 z^4 + 8100246102 z^5 - 13906978176 z^6 + 12302384640 z^7 - 5546311680 z^8 + 1013841920 z^9) EllipticE[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (3568640 + 3011040 z + 9817245 z^2 + 60168525 z^3 - 1394144193 z^4 + 4941079767 z^5 - 7888229856 z^6 + 6626011392 z^7 - 2868203520 z^8 + 506920960 z^9) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + 2 Sqrt[1 - z] (1784320 + 2174640 z + 5907075 z^2 + 32619600 z^3 - 2009671209 z^4 + 8100246102 z^5 - 13906978176 z^6 + 12302384640 z^7 - 5546311680 z^8 + 1013841920 z^9) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + 2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (1784320 + 2174640 z + 5907075 z^2 + 32619600 z^3 - 2009671209 z^4 + 8100246102 z^5 - 13906978176 z^6 + 12302384640 z^7 - 5546311680 z^8 + 1013841920 z^9) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (86875782718725 Pi (1 + Sqrt[1 - z])^(1/4) (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]