Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a97y)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 4 and fixed z

For fixed z and a=-19/4, b>=a

For fixed z and a=-19/4, b=-7/4

http://functions.wolfram.com/07.23.03.a97y.01

Input Form

Hypergeometric2F1[-(19/4), -(7/4), 21/4, z] == (1/(52776558133248 z^(17/4))) (221 (8 (1 - z)^(3/4) z^(1/4) (-855855 + 13535676 z - 115635520 z^2 + 852238464 z^3 + 13536371712 z^4 + 11355004928 z^5 + 640155648 z^6 - 43253760 z^7 + 2097152 z^8) - 131670 Sqrt[2] (13 - 216 z + 1920 z^2 - 14336 z^3 + 258048 z^4 + 688128 z^5 + 262144 z^6) ArcTan[1 - z^(1/4)/(Sqrt[2] (1 - z)^(1/4)), -(z^(1/4)/(Sqrt[2] (1 - z)^(1/4)))] - 131670 Sqrt[2] (13 - 216 z + 1920 z^2 - 14336 z^3 + 258048 z^4 + 688128 z^5 + 262144 z^6) ArcTan[1 + z^(1/4)/(Sqrt[2] (1 - z)^(1/4)), -(z^(1/4)/(Sqrt[2] (1 - z)^(1/4)))] - 65835 Sqrt[2] (13 - 216 z + 1920 z^2 - 14336 z^3 + 258048 z^4 + 688128 z^5 + 262144 z^6) Log[1 - (Sqrt[2] z^(1/4))/(1 - z)^(1/4) + Sqrt[z]/Sqrt[1 - z]] + 65835 Sqrt[2] (13 - 216 z + 1920 z^2 - 14336 z^3 + 258048 z^4 + 688128 z^5 + 262144 z^6) Log[1 + (Sqrt[2] z^(1/4))/(1 - z)^(1/4) + Sqrt[z]/Sqrt[1 - z]]))

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]