Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b7ek)

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=-43/8, b>=a

For fixed z and a=-43/8, b=-39/8

http://functions.wolfram.com/07.23.03.b7ek.01

Input Form

Hypergeometric2F1[-(43/8), -(39/8), 4, z] == (2048 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (2907944832 - 94939855101 z + 2400213120669 z^2 + 157985857100275 z^3 + 737602147347045 z^4 + 925891544086593 z^5 + 356154108606191 z^6 + 35793444985161 z^7 + 423987371535 z^8) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (2907944832 - 94939855101 z + 2400213120669 z^2 + 157985857100275 z^3 + 737602147347045 z^4 + 925891544086593 z^5 + 356154108606191 z^6 + 35793444985161 z^7 + 423987371535 z^8) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[1 - z] (2907944832 - 94939855101 z + 2400213120669 z^2 + 157985857100275 z^3 + 737602147347045 z^4 + 925891544086593 z^5 + 356154108606191 z^6 + 35793444985161 z^7 + 423987371535 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (-2907944832 + 96030334413 z - 2435517388395 z^2 + 85707098810129 z^3 + 893460409109865 z^4 + 1932059683550727 z^5 + 1267064812298047 z^6 + 246483271460955 z^7 + 9885661183491 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (248629711489904745 Pi (1 + Sqrt[1 - z])^(1/4) z^3)

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]