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

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=-39/8, b>=a

For fixed z and a=-39/8, b=37/8

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

Input Form

Hypergeometric2F1[-(39/8), 37/8, 5, z] == (65536 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (-2190336 - 4768544 z - 14748405 z^2 - 83399610 z^3 + 1330226975 z^4 - 3725279712 z^5 + 4475826432 z^6 - 2525614080 z^7 + 551485440 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) (-2190336 - 4768544 z - 14748405 z^2 - 83399610 z^3 + 1330226975 z^4 - 3725279712 z^5 + 4475826432 z^6 - 2525614080 z^7 + 551485440 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[1 - z] (-2190336 - 4768544 z - 14748405 z^2 - 83399610 z^3 + 1330226975 z^4 - 3725279712 z^5 + 4475826432 z^6 - 2525614080 z^7 + 551485440 z^8) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 (-1095168 - 1973584 z - 6367803 z^2 - 38630340 z^3 - 978210695 z^4 + 5507463654 z^5 - 11260706688 z^6 + 11254746624 z^7 - 5579735040 z^8 + 1102970880 z^9) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/(108782333334675 Pi (1 + Sqrt[1 - z])^(1/4) z^4)

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]