Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b6ko)

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=1/8

http://functions.wolfram.com/07.23.03.b6ko.01

Input Form

Hypergeometric2F1[-(45/8), 1/8, 5, z] == (65536 2^(1/4) (-4 Sqrt[1 - z] (64276992 - 763958832 z + 4543621719 z^2 - 21209524245 z^3 - 51576345170 z^4 + 20988482606 z^5 - 8796592581 z^6 + 2757826247 z^7 - 544617920 z^8 + 50069712 z^9) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - 2 Sqrt[2 - 2 Sqrt[1 - z]] Sqrt[1 - z] (64276992 - 763958832 z + 4543621719 z^2 - 21209524245 z^3 - 51576345170 z^4 + 20988482606 z^5 - 8796592581 z^6 + 2757826247 z^7 - 544617920 z^8 + 50069712 z^9) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + (128553984 - 1608263904 z + 10029010173 z^2 - 47948083365 z^3 + 469913100230 z^4 + 7761813542 z^5 - 3183344647 z^6 + 971122399 z^7 - 186506540 z^8 + 16689904 z^9) EllipticK[ 2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] + 2 Sqrt[1 - z] (64276992 - 763958832 z + 4543621719 z^2 - 21209524245 z^3 - 51576345170 z^4 + 20988482606 z^5 - 8796592581 z^6 + 2757826247 z^7 - 544617920 z^8 + 50069712 z^9) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]))/ (17937727563921165 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]] 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]