Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b4rh)

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.b4rh.01

Input Form

Hypergeometric2F1[-(47/8), -(27/8), 5, z] == (65536 2^(1/4) (-4 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (38141415936 - 913407450384 z + 12800989238661 z^2 - 180679165066764 z^3 - 6568386903642529 z^4 - 17983091280748566 z^5 - 12159607650779421 z^6 - 1899854395969616 z^7 - 3176799289191 z^8 + 83234915874 z^9) EllipticE[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (38141415936 - 913407450384 z + 12800989238661 z^2 - 180679165066764 z^3 - 6568386903642529 z^4 - 17983091280748566 z^5 - 12159607650779421 z^6 - 1899854395969616 z^7 - 3176799289191 z^8 + 83234915874 z^9) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 2 Sqrt[1 - z] (38141415936 - 913407450384 z + 12800989238661 z^2 - 180679165066764 z^3 - 6568386903642529 z^4 - 17983091280748566 z^5 - 12159607650779421 z^6 - 1899854395969616 z^7 - 3176799289191 z^8 + 83234915874 z^9) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (76282831872 - 1855420962720 z + 26279212095045 z^2 - 370775907596835 z^3 + 9228372963448345 z^4 + 55423374043937361 z^5 + 68052890962916895 z^6 + 21684379307893655 z^7 + 1113475087104435 z^8 - 25733461491045 z^9 + 665879326992 z^10) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (728503609121502239025 Pi (1 + Sqrt[1 - z])^(1/4) z^4)

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]