Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b8zl)

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=3/8

http://functions.wolfram.com/07.23.03.b8zl.01

Input Form

Hypergeometric2F1[-(41/8), 3/8, 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-1256161280 + 14607781760 z - 79969991805 z^2 + 282356250825 z^3 - 814257250950 z^4 - 5059614394926 z^5 + 1833343893927 z^6 - 728332514091 z^7 + 215777359980 z^8 - 40244538880 z^9 + 3499525120 z^10) EllipticE[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (-1256161280 + 15078842240 z - 85319104365 z^2 + 310915557480 z^3 - 912691755525 z^4 + 330020176824 z^5 + 488200451193 z^6 - 191350596528 z^7 + 55752883641 z^8 - 10225174960 z^9 + 874881280 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (-1256161280 + 14607781760 z - 79969991805 z^2 + 282356250825 z^3 - 814257250950 z^4 - 5059614394926 z^5 + 1833343893927 z^6 - 728332514091 z^7 + 215777359980 z^8 - 40244538880 z^9 + 3499525120 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 + 14607781760 z - 79969991805 z^2 + 282356250825 z^3 - 814257250950 z^4 - 5059614394926 z^5 + 1833343893927 z^6 - 728332514091 z^7 + 215777359980 z^8 - 40244538880 z^9 + 3499525120 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (1339391376632283975 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]