Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.bc2c)

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=-37/8, b>=a

For fixed z and a=-37/8, b=31/8

http://functions.wolfram.com/07.23.03.bc2c.01

Input Form

Hypergeometric2F1[-(37/8), 31/8, 3, z] == (256 2^(1/4) (8 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-195286 - 2441075 z + 70661415 z^2 - 278168460 z^3 + 433451040 z^4 - 302455296 z^5 + 79073280 z^6) EllipticE[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - 4 Sqrt[1 - z] (-195286 - 2441075 z + 70661415 z^2 - 278168460 z^3 + 433451040 z^4 - 302455296 z^5 + 79073280 z^6) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - 4 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-195286 - 2441075 z + 70661415 z^2 - 278168460 z^3 + 433451040 z^4 - 302455296 z^5 + 79073280 z^6) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - (-781144 - 9471371 z - 341293110 z^2 + 2738072505 z^3 - 7325692320 z^4 + 9119506176 z^5 - 5445513216 z^6 + 1265172480 z^7) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (80343101475 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) z^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]