Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.ag9r)

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=-5/4, b>=a

For fixed z and a=-5/4, b=19/4

http://functions.wolfram.com/07.23.03.ag9r.01

Input Form

Hypergeometric2F1[-(5/4), 19/4, -(11/2), z] == (1/(40981248 Pi^(3/2))) (((1/(-1 + z)^9) (2 (-10245312 + 79866864 z - 261578856 z^2 + 450137919 z^3 - 388785320 z^4 + 57891834 z^5 + 115783668 z^6 - 191977617 z^7 + 125489988 z^8 - 40734720 z^9 + 5431296 z^10) EllipticE[(1/2) (1 - Sqrt[z])]) + (1/(-1 + z)^9) (2 (-10245312 + 79866864 z - 261578856 z^2 + 450137919 z^3 - 388785320 z^4 + 57891834 z^5 + 115783668 z^6 - 191977617 z^7 + 125489988 z^8 - 40734720 z^9 + 5431296 z^10) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^9 (1 + Sqrt[z])^8)) ((10245312 - 5122656 Sqrt[z] - 74744208 z + 35237664 z^(3/2) + 226341192 z^2 - 98666106 z^(5/2) - 351471813 z^3 + 135776564 z^(7/2) + 253008756 z^4 - 72900828 z^(9/2) + 15008994 z^5 - 34306272 z^(11/2) - 81477396 z^6 + 122062278 z^(13/2) + 69915339 z^7 - 98630532 z^(15/2) - 26859456 z^8 + 36661248 z^(17/2) + 4073472 z^9 - 5431296 z^(19/2)) EllipticK[(1/2) (1 - Sqrt[z])]) - (1/((-1 + Sqrt[z])^8 (1 + Sqrt[z])^9)) ((10245312 + 5122656 Sqrt[z] - 74744208 z - 35237664 z^(3/2) + 226341192 z^2 + 98666106 z^(5/2) - 351471813 z^3 - 135776564 z^(7/2) + 253008756 z^4 + 72900828 z^(9/2) + 15008994 z^5 + 34306272 z^(11/2) - 81477396 z^6 - 122062278 z^(13/2) + 69915339 z^7 + 98630532 z^(15/2) - 26859456 z^8 - 36661248 z^(17/2) + 4073472 z^9 + 5431296 z^(19/2)) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[1/4]^2)

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]