Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b7gm)

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=-43/8, b>=a

For fixed z and a=-43/8, b=-31/8

http://functions.wolfram.com/07.23.03.b7gm.01

Input Form

Hypergeometric2F1[-(43/8), -(31/8), 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (-57264144384 + 1299403963776 z - 15799540579137 z^2 + 150599920145115 z^3 - 1638185609485425 z^4 - 47939978437443321 z^5 - 115673672244356911 z^6 - 74261625241778543 z^7 - 12924341216425755 z^8 - 343484128298675 z^9 + 3087497782460 z^10) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (-57264144384 + 1320878017920 z - 16280945253873 z^2 + 156394627597992 z^3 - 1693109251481340 z^4 + 36497694298718424 z^5 + 186945602639559290 z^6 + 210370829300571928 z^7 + 67923990720561348 z^8 + 5097156842648680 z^9 + 771874445615 z^10) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (-57264144384 + 1299403963776 z - 15799540579137 z^2 + 150599920145115 z^3 - 1638185609485425 z^4 - 47939978437443321 z^5 - 115673672244356911 z^6 - 74261625241778543 z^7 - 12924341216425755 z^8 - 343484128298675 z^9 + 3087497782460 z^10) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + Sqrt[1 - z] (-57264144384 + 1299403963776 z - 15799540579137 z^2 + 150599920145115 z^3 - 1638185609485425 z^4 - 47939978437443321 z^5 - 115673672244356911 z^6 - 74261625241778543 z^7 - 12924341216425755 z^8 - 343484128298675 z^9 + 3087497782460 z^10) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (21977623347150129934275 Pi (1 + Sqrt[1 - z])^(1/4) 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]