Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b8o1)

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=-41/8, b>=a

For fixed z and a=-41/8, b=-37/8

http://functions.wolfram.com/07.23.03.b8o1.01

Input Form

Hypergeometric2F1[-(41/8), -(37/8), 2, z] == (16 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (534734915 + 224906097246 z + 2254310673189 z^2 + 4914771392132 z^3 + 2924457431805 z^4 + 421973658750 z^5 + 6810111035 z^6) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[1 - z] (534734915 + 224906097246 z + 2254310673189 z^2 + 4914771392132 z^3 + 2924457431805 z^4 + 421973658750 z^5 + 6810111035 z^6) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (534734915 + 224906097246 z + 2254310673189 z^2 + 4914771392132 z^3 + 2924457431805 z^4 + 421973658750 z^5 + 6810111035 z^6) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (-534734915 - 77319260706 z - 471519112953 z^2 - 281522216588 z^3 + 519701707707 z^4 + 289750382670 z^5 + 21443234785 z^6) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (1179090487575 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) 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]