Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.an1e)

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=4, b>=a

For fixed z and a=4, b=6

http://functions.wolfram.com/07.23.03.an1e.01

Input Form

Hypergeometric2F1[4, 6, -(19/4), z] == (1/1753219072) (-((1/(-1 + z)^14) (8 (-219152384 + 4175429632 z - 40612397056 z^2 + 282943553536 z^3 - 1784148918272 z^4 + 15724565233664 z^5 + 102814419751225 z^6 + 94818616337512 z^7 + 18550670232944 z^8 + 254754694656 z^9 - 4806692352 z^10))) + (1/(1 - z)^(59/4)) (157437950670 Sqrt[2] z^(23/4) (1953 + 4340 z + 2240 z^2 + 256 z^3) ArcTan[1 - z^(1/4)/(Sqrt[2] (1 - z)^(1/4)), -(z^(1/4)/(Sqrt[2] (1 - z)^(1/4)))]) + (1/(1 - z)^(59/4)) (157437950670 Sqrt[2] z^(23/4) (1953 + 4340 z + 2240 z^2 + 256 z^3) ArcTan[1 + z^(1/4)/(Sqrt[2] (1 - z)^(1/4)), -(z^(1/4)/(Sqrt[2] (1 - z)^(1/4)))]) + (78718975335 Sqrt[2] z^(23/4) (1953 + 4340 z + 2240 z^2 + 256 z^3) Log[1 - (Sqrt[2] z^(1/4))/(1 - z)^(1/4) + Sqrt[z]/Sqrt[1 - z]])/ (1 - z)^(59/4) - (78718975335 Sqrt[2] z^(23/4) (1953 + 4340 z + 2240 z^2 + 256 z^3) Log[1 + (Sqrt[2] z^(1/4))/(1 - z)^(1/4) + Sqrt[z]/Sqrt[1 - z]])/ (1 - z)^(59/4))

Standard Form

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MathML Form

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</semantics> </math>

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]