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

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=37/8

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

Input Form

Hypergeometric2F1[-(33/8), 37/8, 6, z] == (1/(4022196326223075 Pi z^5)) (262144 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (4 (91914240 - 74321280 z - 68568225 z^2 - 132028545 z^3 - 523589715 z^4 + 20131660957 z^5 - 56669840344 z^6 + 65186381040 z^7 - 34867901376 z^8 + 7210038528 z^9) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 20 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (2872320 - 1851300 z - 2302905 z^2 - 4521660 z^3 + 749953215 z^4 - 2232912034 z^5 + 2637478540 z^6 - 1434947184 z^7 + 300418272 z^8) EllipticK[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (-11489280 + 942480 z + 8445855 z^2 + 22310970 z^3 + 5196528975 z^4 - 16400104904 z^5 + 20073032144 z^6 - 11222076096 z^7 + 2403346176 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 2 (91914240 - 74321280 z - 68568225 z^2 - 132028545 z^3 - 523589715 z^4 + 20131660957 z^5 - 56669840344 z^6 + 65186381040 z^7 - 34867901376 z^8 + 7210038528 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))

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]