Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b9b7)

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.b9b7.01

Input Form

Hypergeometric2F1[-(41/8), 27/8, 4, z] == (2048 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (1338240 + 2143275 z + 10193625 z^2 - 584989379 z^3 + 2276280383 z^4 - 3819999456 z^5 + 3325620480 z^6 - 1481625600 z^7 + 268369920 z^8) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (1338240 + 1641435 z + 9252675 z^2 - 201709079 z^3 + 692144297 z^4 - 1081654704 z^5 + 894880896 z^6 - 382986240 z^7 + 67092480 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (1338240 + 2143275 z + 10193625 z^2 - 584989379 z^3 + 2276280383 z^4 - 3819999456 z^5 + 3325620480 z^6 - 1481625600 z^7 + 268369920 z^8) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (1338240 + 2143275 z + 10193625 z^2 - 584989379 z^3 + 2276280383 z^4 - 3819999456 z^5 + 3325620480 z^6 - 1481625600 z^7 + 268369920 z^8) EllipticK[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (396693071775 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]