Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a90m)

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=-19/4

http://functions.wolfram.com/07.23.03.a90m.01

Input Form

Hypergeometric2F1[-(19/4), -(19/4), 21/4, z] == (221 (8 (1 - z)^(3/4) z^(1/4) (-894368475 + 21574919520 z - 299822889520 z^2 + 3904033056000 z^3 + 174934599135744 z^4 + 509351048413184 z^5 + 383142549356544 z^6 + 78970074169344 z^7 + 3078628573184 z^8) - 131670 Sqrt[2] (13585 - 338580 z + 4815360 z^2 - 62920704 z^3 + 2265145344 z^4 + 15100968960 z^5 + 23011000320 z^6 + 10758389760 z^7 + 1434451968 z^8 + 33554432 z^9) ArcTan[1 - z^(1/4)/(Sqrt[2] (1 - z)^(1/4)), -(z^(1/4)/(Sqrt[2] (1 - z)^(1/4)))] - 131670 Sqrt[2] (13585 - 338580 z + 4815360 z^2 - 62920704 z^3 + 2265145344 z^4 + 15100968960 z^5 + 23011000320 z^6 + 10758389760 z^7 + 1434451968 z^8 + 33554432 z^9) ArcTan[1 + z^(1/4)/(Sqrt[2] (1 - z)^(1/4)), -(z^(1/4)/(Sqrt[2] (1 - z)^(1/4)))] - 65835 Sqrt[2] (13585 - 338580 z + 4815360 z^2 - 62920704 z^3 + 2265145344 z^4 + 15100968960 z^5 + 23011000320 z^6 + 10758389760 z^7 + 1434451968 z^8 + 33554432 z^9) Log[1 - (Sqrt[2] z^(1/4))/(1 - z)^(1/4) + Sqrt[z]/Sqrt[1 - z]] + 65835 Sqrt[2] (13585 - 338580 z + 4815360 z^2 - 62920704 z^3 + 2265145344 z^4 + 15100968960 z^5 + 23011000320 z^6 + 10758389760 z^7 + 1434451968 z^8 + 33554432 z^9) Log[1 + (Sqrt[2] z^(1/4))/(1 - z)^(1/4) + Sqrt[z]/Sqrt[1 - z]]))/ (567453553048682496 z^(17/4))

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]