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

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=-31/8, b>=a

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

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

Input Form

Hypergeometric2F1[-(31/8), 37/8, 5, z] == (65536 2^(1/4) (-8 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (182528 + 359352 z + 973245 z^2 + 4666585 z^3 - 56673540 z^4 + 117776736 z^5 - 93789696 z^6 + 26357760 z^7) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 4 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (182528 + 359352 z + 973245 z^2 + 4666585 z^3 - 56673540 z^4 + 117776736 z^5 - 93789696 z^6 + 26357760 z^7) EllipticK[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 4 Sqrt[1 - z] (182528 + 359352 z + 973245 z^2 + 4666585 z^3 - 56673540 z^4 + 117776736 z^5 - 93789696 z^6 + 26357760 z^7) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (730112 + 1163616 z + 3279087 z^2 + 17019310 z^3 + 373648275 z^4 - 1672314336 z^5 + 2558637312 z^6 - 1702711296 z^7 + 421724160 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (19917892019025 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]