Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b4r9)

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=-47/8, b>=a

For fixed z and a=-47/8, b=-27/8

http://functions.wolfram.com/07.23.03.b4r9.01

Input Form

Hypergeometric2F1[-(47/8), -(27/8), 3, z] == (256 2^(1/4) (-8 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (49663302 - 2085858684 z - 167246553313 z^2 - 761146077195 z^3 - 768331409580 z^4 - 167545031402 z^5 - 419540121 z^6 + 13830993 z^7) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 4 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (49663302 - 2085858684 z - 167246553313 z^2 - 761146077195 z^3 - 768331409580 z^4 - 167545031402 z^5 - 419540121 z^6 + 13830993 z^7) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 4 Sqrt[1 - z] (49663302 - 2085858684 z - 167246553313 z^2 - 761146077195 z^3 - 768331409580 z^4 - 167545031402 z^5 - 419540121 z^6 + 13830993 z^7) EllipticK[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (198653208 - 8417929689 z + 345784036514 z^2 + 3814546564437 z^3 + 7262184490260 z^4 + 3297436827937 z^5 + 228552548994 z^6 - 6818679549 z^7 + 221295888 z^8) EllipticK[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (129492714426075 Pi (1 + Sqrt[1 - z])^(1/4) z^2)

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]