Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.be7x)

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=-33/8, b>=a

For fixed z and a=-33/8, b=19/8

http://functions.wolfram.com/07.23.03.be7x.01

Input Form

Hypergeometric2F1[-(33/8), 19/8, 5, z] == (65536 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-783360 + 3386400 z - 3596775 z^2 - 3132675 z^3 + 57847595 z^4 - 102253697 z^5 + 83297760 z^6 - 33895680 z^7 + 5591040 z^8) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - 2 (-391680 + 1840080 z - 2393175 z^2 - 1044225 z^3 + 8909485 z^4 - 14395771 z^5 + 11144406 z^6 - 4368000 z^7 + 698880 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[1 - z] (-783360 + 3386400 z - 3596775 z^2 - 3132675 z^3 + 57847595 z^4 - 102253697 z^5 + 83297760 z^6 - 33895680 z^7 + 5591040 z^8) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-783360 + 3386400 z - 3596775 z^2 - 3132675 z^3 + 57847595 z^4 - 102253697 z^5 + 83297760 z^6 - 33895680 z^7 + 5591040 z^8) EllipticK[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (1357107877125 Pi (1 + Sqrt[1 - z])^(1/4) (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]