Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b6sd)

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=15/8

http://functions.wolfram.com/07.23.03.b6sd.01

Input Form

Hypergeometric2F1[-(45/8), 15/8, 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (1371242496 - 11275255680 z + 37960599365 z^2 - 60093059195 z^3 + 11204534250 z^4 - 37224262670 z^5 + 47101387025 z^6 - 34798591575 z^7 + 15661773600 z^8 - 4003296000 z^9 + 448081920 z^10) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (1371242496 - 11275255680 z + 37960599365 z^2 - 60093059195 z^3 + 11204534250 z^4 - 37224262670 z^5 + 47101387025 z^6 - 34798591575 z^7 + 15661773600 z^8 - 4003296000 z^9 + 448081920 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (1371242496 - 11275255680 z + 37960599365 z^2 - 60093059195 z^3 + 11204534250 z^4 - 37224262670 z^5 + 47101387025 z^6 - 34798591575 z^7 + 15661773600 z^8 - 4003296000 z^9 + 448081920 z^10) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (1371242496 - 11789471616 z + 42048214325 z^2 - 73246826705 z^3 + 30412307250 z^4 + 101711962030 z^5 - 227176158055 z^6 + 249223057875 z^7 - 167604135600 z^8 + 70129488000 z^9 - 16872007680 z^10 + 1792327680 z^11) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (36353459088171225 Pi (1 + Sqrt[1 - z])^(1/4) (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]