Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b97a)

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=-41/8, b>=a

For fixed z and a=-41/8, b=19/8

http://functions.wolfram.com/07.23.03.b97a.01

Input Form

Hypergeometric2F1[-(41/8), 19/8, 6, z] == (524288 2^(1/4) (-2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-114196480 + 749860480 z - 1820016855 z^2 + 1465163700 z^3 + 1132171950 z^4 - 19940229036 z^5 + 36809511921 z^6 - 34340853456 z^7 + 18371283840 z^8 - 5394421760 z^9 + 678379520 z^10) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (-114196480 + 792684160 z - 2089504935 z^2 + 2077000755 z^3 + 732581850 z^4 - 6154370586 z^5 + 10471785597 z^6 - 9335750697 z^7 + 4830579936 z^8 - 1380404480 z^9 + 169594880 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[1 - z] (-114196480 + 749860480 z - 1820016855 z^2 + 1465163700 z^3 + 1132171950 z^4 - 19940229036 z^5 + 36809511921 z^6 - 34340853456 z^7 + 18371283840 z^8 - 5394421760 z^9 + 678379520 z^10) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-114196480 + 749860480 z - 1820016855 z^2 + 1465163700 z^3 + 1132171950 z^4 - 19940229036 z^5 + 36809511921 z^6 - 34340853456 z^7 + 18371283840 z^8 - 5394421760 z^9 + 678379520 z^10) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (3744803476138725 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]