Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b6p2)

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=-45/8, b>=a

For fixed z and a=-45/8, b=9/8

http://functions.wolfram.com/07.23.03.b6p2.01

Input Form

Hypergeometric2F1[-(45/8), 9/8, 6, z] == (524288 2^(1/4) (-2 Sqrt[1 - z] (4113727488 - 40092773760 z + 173840179695 z^2 - 435753857175 z^3 + 657903887550 z^4 - 988947015406 z^5 + 883578316075 z^6 - 523970263515 z^7 + 201227851000 z^8 - 45479988400 z^9 + 4606413504 z^10) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (4113727488 - 40092773760 z + 173840179695 z^2 - 435753857175 z^3 + 657903887550 z^4 - 988947015406 z^5 + 883578316075 z^6 - 523970263515 z^7 + 201227851000 z^8 - 45479988400 z^9 + 4606413504 z^10) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + (4113727488 - 42663853440 z + 198476345535 z^2 - 540490621125 z^3 + 914231409000 z^4 - 199462350346 z^5 + 169414828415 z^6 - 96378167185 z^7 + 35693528650 z^8 - 7808490800 z^9 + 767735584 z^10) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + Sqrt[1 - z] (4113727488 - 40092773760 z + 173840179695 z^2 - 435753857175 z^3 + 657903887550 z^4 - 988947015406 z^5 + 883578316075 z^6 - 523970263515 z^7 + 201227851000 z^8 - 45479988400 z^9 + 4606413504 z^10) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/ (304941368586659805 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]] z^5)

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]