Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b6bn)

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=-45/8, b>=a

For fixed z and a=-45/8, b=-23/8

http://functions.wolfram.com/07.23.03.b6bn.01

Input Form

Hypergeometric2F1[-(45/8), -(23/8), 4, z] == (2048 2^(1/4) (2 Sqrt[1 - z] (14421120 - 357486045 z + 6596986410 z^2 + 204448361293 z^3 + 483953655680 z^4 + 232142888685 z^5 + 12566289890 z^6 - 536008445 z^7 + 19704468 z^8) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (14421120 - 357486045 z + 6596986410 z^2 + 204448361293 z^3 + 483953655680 z^4 + 232142888685 z^5 + 12566289890 z^6 - 536008445 z^7 + 19704468 z^8) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - (14421120 - 366499245 z + 6818936460 z^2 - 264525109127 z^3 - 1502215564010 z^4 - 1625987739995 z^5 - 369046669360 z^6 - 90312145 z^7 + 3284078 z^8) EllipticK[ 2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - Sqrt[1 - z] (14421120 - 357486045 z + 6596986410 z^2 + 204448361293 z^3 + 483953655680 z^4 + 232142888685 z^5 + 12566289890 z^6 - 536008445 z^7 + 19704468 z^8) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/(476043588652905 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]] 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]