Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b8x1)

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=-5/8

http://functions.wolfram.com/07.23.03.b8x1.01

Input Form

Hypergeometric2F1[-(41/8), -(5/8), 6, z] == (524288 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (1256161280 - 17787440000 z + 124452085725 z^2 - 608338498350 z^3 + 2843426746875 z^4 + 54431326077804 z^5 + 20774441195235 z^6 - 2740347865710 z^7 + 486263604645 z^8 - 64717780400 z^9 + 4374406400 z^10) EllipticE[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - (1256161280 - 18258500480 z + 130993570125 z^2 - 653252551050 z^3 + 3059716650375 z^4 + 7436888130204 z^5 - 9360053906229 z^6 - 706500808650 z^7 + 124502148225 z^8 - 16384495400 z^9 + 1093601600 z^10) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[1 - z] (1256161280 - 17787440000 z + 124452085725 z^2 - 608338498350 z^3 + 2843426746875 z^4 + 54431326077804 z^5 + 20774441195235 z^6 - 2740347865710 z^7 + 486263604645 z^8 - 64717780400 z^9 + 4374406400 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (1256161280 - 17787440000 z + 124452085725 z^2 - 608338498350 z^3 + 2843426746875 z^4 + 54431326077804 z^5 + 20774441195235 z^6 - 2740347865710 z^7 + 486263604645 z^8 - 64717780400 z^9 + 4374406400 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (12054522389690555775 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]