Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b8wt)

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.b8wt.01

Input Form

Hypergeometric2F1[-(41/8), -(5/8), 4, z] == (2048 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (14720640 - 215634375 z + 2010862425 z^2 + 66627684994 z^3 + 38100166650 z^4 - 6985467307 z^5 + 1645139405 z^6 - 280578480 z^7 + 23645440 z^8) EllipticE[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - (14720640 - 221154615 z + 2090215875 z^2 + 12087713794 z^3 - 12512498154 z^4 - 1817566387 z^5 + 423910487 z^6 - 71253000 z^7 + 5911360 z^8) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[1 - z] (14720640 - 215634375 z + 2010862425 z^2 + 66627684994 z^3 + 38100166650 z^4 - 6985467307 z^5 + 1645139405 z^6 - 280578480 z^7 + 23645440 z^8) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (14720640 - 215634375 z + 2010862425 z^2 + 66627684994 z^3 + 38100166650 z^4 - 6985467307 z^5 + 1645139405 z^6 - 280578480 z^7 + 23645440 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (55098579811275 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) z^3)

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]