Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.bbb3)

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=-37/8, b>=a

For fixed z and a=-37/8, b=-25/8

http://functions.wolfram.com/07.23.03.bbb3.01

Input Form

Hypergeometric2F1[-(37/8), -(25/8), 6, z] == (524288 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (2988605440 - 58196086400 z + 593988523375 z^2 - 4613487726575 z^3 + 39196490180575 z^4 + 1926764355569977 z^5 + 3924210283681765 z^6 + 1855041869805355 z^7 + 185731604081485 z^8 + 19501972875 z^9) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[1 - z] (2988605440 - 58196086400 z + 593988523375 z^2 - 4613487726575 z^3 + 39196490180575 z^4 + 1926764355569977 z^5 + 3924210283681765 z^6 + 1855041869805355 z^7 + 185731604081485 z^8 + 19501972875 z^9) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (2988605440 - 58196086400 z + 593988523375 z^2 - 4613487726575 z^3 + 39196490180575 z^4 + 1926764355569977 z^5 + 3924210283681765 z^6 + 1855041869805355 z^7 + 185731604081485 z^8 + 19501972875 z^9) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - (2988605440 - 59316813440 z + 615505606975 z^2 - 4830428851625 z^3 + 40868704669975 z^4 + 444575032205227 z^5 + 143514343574833 z^6 - 449036138410475 z^7 - 168766494162035 z^8 - 6962204316375 z^9 + 78007891500 z^10) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/(384809918203371986325 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) z^5)

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]