Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b976)

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=19/8

http://functions.wolfram.com/07.23.03.b976.01

Input Form

Hypergeometric2F1[-(41/8), 19/8, 5, z] == (65536 2^(1/4) (-16 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (4014720 - 19529940 z + 23785125 z^2 + 23785125 z^3 - 537561895 z^4 + 1189120439 z^5 - 1292201638 z^6 + 788970000 z^7 - 260332800 z^8 + 36341760 z^9) EllipticE[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + 8 Sqrt[1 - z] (4014720 - 19529940 z + 23785125 z^2 + 23785125 z^3 - 537561895 z^4 + 1189120439 z^5 - 1292201638 z^6 + 788970000 z^7 - 260332800 z^8 + 36341760 z^9) EllipticK[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + 8 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (4014720 - 19529940 z + 23785125 z^2 + 23785125 z^3 - 537561895 z^4 + 1189120439 z^5 - 1292201638 z^6 + 788970000 z^7 - 260332800 z^8 + 36341760 z^9) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (32117760 - 168283680 z + 245577495 z^2 + 133196700 z^3 - 1360653710 z^4 + 2748528172 z^5 - 2837963921 z^6 + 1669091424 z^7 - 534293760 z^8 + 72683520 z^9) EllipticK[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (99068875030125 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]