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

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 8 and fixed z and a<0

For fixed z and a=-33/8, b>=a

For fixed z and a=-33/8, b=45/8

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

Input Form

Hypergeometric2F1[-(33/8), 45/8, 6, z] == (262144 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (4 (-275742720 - 406074240 z - 888559485 z^2 - 2747261880 z^3 - 15117168825 z^4 + 760461042334 z^5 - 2599343130192 z^6 + 3515097291552 z^7 - 2160608212224 z^8 + 504702696960 z^9) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (-34467840 - 56414160 z - 122048355 z^2 - 367146450 z^3 + 111583598445 z^4 - 406466240152 z^5 + 566614976112 z^6 - 355094397504 z^7 + 84117116160 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 6 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (17233920 + 37901160 z + 84287445 z^2 + 236209050 z^3 + 95742069225 z^4 - 369723304624 z^5 + 535679697168 z^6 - 346081849344 z^7 + 84117116160 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 (-275742720 - 406074240 z - 888559485 z^2 - 2747261880 z^3 - 15117168825 z^4 + 760461042334 z^5 - 2599343130192 z^6 + 3515097291552 z^7 - 2160608212224 z^8 + 504702696960 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (148821264070253775 Pi z^5)

Standard Form

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MathML Form

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</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]