Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b9fi)

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=35/8

http://functions.wolfram.com/07.23.03.b9fi.01

Input Form

Hypergeometric2F1[-(41/8), 35/8, 4, z] == (2048 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (446080 + 2526625 z + 21293350 z^2 - 1798913987 z^3 + 9049269580 z^4 - 18611220992 z^5 + 19179596800 z^6 - 9870049280 z^7 + 2027683840 z^8) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (446080 + 2359345 z + 20300125 z^2 - 644995637 z^3 + 2825581447 z^4 - 5360462432 z^5 + 5212410112 z^6 - 2562560000 z^7 + 506920960 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (446080 + 2526625 z + 21293350 z^2 - 1798913987 z^3 + 9049269580 z^4 - 18611220992 z^5 + 19179596800 z^6 - 9870049280 z^7 + 2027683840 z^8) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (446080 + 2526625 z + 21293350 z^2 - 1798913987 z^3 + 9049269580 z^4 - 18611220992 z^5 + 19179596800 z^6 - 9870049280 z^7 + 2027683840 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (1190079215325 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]