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

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=5, b>=a

For fixed z and a=5, b=5

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

Input Form

Hypergeometric2F1[5, 5, -(19/4), z] == (1/7012876288) (-((1/(-1 + z)^14) (8 (-876609536 + 16886267904 z - 166509674496 z^2 + 1180512026624 z^3 - 7616222724096 z^4 + 69216208158720 z^5 + 479531528714945 z^6 + 487218472941300 z^7 + 118569549495840 z^8 + 5152241768320 z^9))) + (1/(1 - z)^(59/4)) (3661347690 Sqrt[2] z^(23/4) (380835 + 902720 z + 524160 z^2 + 79872 z^3 + 2048 z^4) 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)) (3661347690 Sqrt[2] z^(23/4) (380835 + 902720 z + 524160 z^2 + 79872 z^3 + 2048 z^4) 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)) (1830673845 Sqrt[2] z^(23/4) (380835 + 902720 z + 524160 z^2 + 79872 z^3 + 2048 z^4) Log[1 - (Sqrt[2] z^(1/4))/(1 - z)^(1/4) + Sqrt[z]/Sqrt[1 - z]]) - (1/(1 - z)^(59/4)) (1830673845 Sqrt[2] z^(23/4) (380835 + 902720 z + 524160 z^2 + 79872 z^3 + 2048 z^4) 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]