Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a90x)

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=-19/4, b>=a

For fixed z and a=-19/4, b=-9/2

http://functions.wolfram.com/07.23.03.a90x.01

Input Form

Hypergeometric2F1[-(19/4), -(9/2), 4, z] == (Sqrt[2] (2 (1 - z)^(1/4) (136192 - 3907008 z + 85379616 z^2 + 4138602560 z^3 + 14818942560 z^4 + 13332152628 z^5 + 3213881118 z^6 + 144827109 z^7) EllipticE[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/ (2 z)] + 2 (1 - z)^(3/4) (136192 - 3907008 z + 85379616 z^2 + 4138602560 z^3 + 14818942560 z^4 + 13332152628 z^5 + 3213881118 z^6 + 144827109 z^7) EllipticE[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/ (2 z)] - (1 - z)^(1/4) (136192 - 3907008 z + 85379616 z^2 + 4138602560 z^3 + 14818942560 z^4 + 13332152628 z^5 + 3213881118 z^6 + 144827109 z^7) EllipticK[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/ (2 z)] - Sqrt[1 - z] (136192 - 3907008 z + 85379616 z^2 + 4138602560 z^3 + 14818942560 z^4 + 13332152628 z^5 + 3213881118 z^6 + 144827109 z^7) EllipticK[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/ (2 z)] - (1 - z)^(3/4) (136192 - 3907008 z + 85379616 z^2 + 4138602560 z^3 + 14818942560 z^4 + 13332152628 z^5 + 3213881118 z^6 + 144827109 z^7) EllipticK[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/ (2 z)] + (-136192 + 3975104 z - 87320352 z^2 - 434924960 z^3 + 6365842240 z^4 + 17545703052 z^5 + 10707839586 z^6 + 1595490057 z^7 + 33546240 z^8) EllipticK[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/ (2 z)]))/(1830673845 Pi Sqrt[1 + Sqrt[1 - z]] z^3)

Standard Form

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MathML Form

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z </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 144827109 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3213881118 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 13332152628 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 14818942560 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4138602560 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 85379616 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 3907008 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mn> 136192 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> 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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]