Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b7ge)

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=-31/8

http://functions.wolfram.com/07.23.03.b7ge.01

Input Form

Hypergeometric2F1[-(43/8), -(31/8), 4, z] == (2048 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (-4697449344 + 133693812189 z - 2886362428950 z^2 - 151915749446665 z^3 - 550559676515680 z^4 - 493756962685861 z^5 - 114302137723214 z^6 - 3902858112335 z^7 + 43485884260 z^8) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (-4697449344 + 135455355693 z - 2936015936469 z^2 + 91955636141165 z^3 + 752008595868275 z^4 + 1212700614084719 z^5 + 528953618086537 z^6 + 51478451709559 z^7 + 10871471065 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) (-4697449344 + 133693812189 z - 2886362428950 z^2 - 151915749446665 z^3 - 550559676515680 z^4 - 493756962685861 z^5 - 114302137723214 z^6 - 3902858112335 z^7 + 43485884260 z^8) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + Sqrt[1 - z] (-4697449344 + 133693812189 z - 2886362428950 z^2 - 151915749446665 z^3 - 550559676515680 z^4 - 493756962685861 z^5 - 114302137723214 z^6 - 3902858112335 z^7 + 43485884260 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]