Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9051)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 4 and fixed z

For fixed z and a=-23/4, b>=a

For fixed z and a=-23/4, b=-9/2

http://functions.wolfram.com/07.23.03.9051.01

Input Form

Hypergeometric2F1[-(23/4), -(9/2), 1, z] == (2 (1 - z)^(1/4) (15733456 + 261582720 z + 745800516 z^2 + 509352116 z^3 + 71542401 z^4 + 193536 z^5) EllipticE[ (2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] + 2 (1 - z)^(3/4) (15733456 + 261582720 z + 745800516 z^2 + 509352116 z^3 + 71542401 z^4 + 193536 z^5) EllipticE[ (2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] - (1 - z)^(1/4) (15733456 + 261582720 z + 745800516 z^2 + 509352116 z^3 + 71542401 z^4 + 193536 z^5) EllipticK[ (2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] - Sqrt[1 - z] (15733456 + 261582720 z + 745800516 z^2 + 509352116 z^3 + 71542401 z^4 + 193536 z^5) EllipticK[ (2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] - (1 - z)^(3/4) (15733456 + 261582720 z + 745800516 z^2 + 509352116 z^3 + 71542401 z^4 + 193536 z^5) EllipticK[ (2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] + (-6042544 - 4175008 z + 396894852 z^2 + 833142976 z^3 + 359031253 z^4 + 25546752 z^5 - 193536 z^6) EllipticK[ (2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)])/ (2422728 Sqrt[2] Pi 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]