Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.bi4g)

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=-27/8, b>=a

For fixed z and a=-27/8, b=33/8

http://functions.wolfram.com/07.23.03.bi4g.01

Input Form

Hypergeometric2F1[-(27/8), 33/8, 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (-61636608 + 92695680 z + 41366325 z^2 + 40422690 z^3 + 79726185 z^4 - 1828807792 z^5 + 3374367360 z^6 - 2334689280 z^7 + 574013440 z^8) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (-61636608 + 115809408 z + 12925605 z^2 + 18762975 z^3 + 57473955 z^4 - 571425127 z^5 + 940249632 z^6 - 610579200 z^7 + 143503360 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (-61636608 + 92695680 z + 41366325 z^2 + 40422690 z^3 + 79726185 z^4 - 1828807792 z^5 + 3374367360 z^6 - 2334689280 z^7 + 574013440 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + Sqrt[1 - z] (-61636608 + 92695680 z + 41366325 z^2 + 40422690 z^3 + 79726185 z^4 - 1828807792 z^5 + 3374367360 z^6 - 2334689280 z^7 + 574013440 z^8) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/(339577769725425 Pi (1 + Sqrt[1 - z])^(1/4) z^5)

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]