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

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=43/8

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

Input Form

Hypergeometric2F1[-(33/8), 43/8, 5, z] == (65536 2^(1/4) (8 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-65280 - 248200 z - 1049750 z^2 - 7176975 z^3 + 483169115 z^4 - 1909963328 z^5 + 2911184640 z^6 - 1982955520 z^7 + 506920960 z^8) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - (-261120 - 894880 z - 3799925 z^2 - 27017250 z^3 + 672794135 z^4 - 2307579872 z^5 + 3237841152 z^6 - 2078003200 z^7 + 506920960 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - 4 Sqrt[1 - z] (-65280 - 248200 z - 1049750 z^2 - 7176975 z^3 + 483169115 z^4 - 1909963328 z^5 + 2911184640 z^6 - 1982955520 z^7 + 506920960 z^8) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - 4 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-65280 - 248200 z - 1049750 z^2 - 7176975 z^3 + 483169115 z^4 - 1909963328 z^5 + 2911184640 z^6 - 1982955520 z^7 + 506920960 z^8) EllipticK[ 1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (41652772536375 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]