Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.8971)

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=-11/2

http://functions.wolfram.com/07.23.03.8971.01

Input Form

Hypergeometric2F1[-(23/4), -(11/2), 4, z] == (2 (1 - z)^(1/4) (9844736 - 362409344 z + 10504948608 z^2 + 757444288960 z^3 + 4189539125600 z^4 + 6538595210184 z^5 + 3375456537024 z^6 + 531937693782 z^7 + 16928279775 z^8) EllipticE[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] + 2 (1 - z)^(3/4) (9844736 - 362409344 z + 10504948608 z^2 + 757444288960 z^3 + 4189539125600 z^4 + 6538595210184 z^5 + 3375456537024 z^6 + 531937693782 z^7 + 16928279775 z^8) EllipticE[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] - (1 - z)^(1/4) (9844736 - 362409344 z + 10504948608 z^2 + 757444288960 z^3 + 4189539125600 z^4 + 6538595210184 z^5 + 3375456537024 z^6 + 531937693782 z^7 + 16928279775 z^8) EllipticK[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] - Sqrt[1 - z] (9844736 - 362409344 z + 10504948608 z^2 + 757444288960 z^3 + 4189539125600 z^4 + 6538595210184 z^5 + 3375456537024 z^6 + 531937693782 z^7 + 16928279775 z^8) EllipticK[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] - (1 - z)^(3/4) (9844736 - 362409344 z + 10504948608 z^2 + 757444288960 z^3 + 4189539125600 z^4 + 6538595210184 z^5 + 3375456537024 z^6 + 531937693782 z^7 + 16928279775 z^8) EllipticK[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)] + (-9844736 + 367331712 z - 10685230336 z^2 - 129796221760 z^3 + 1033427654880 z^4 + 5504595327736 z^5 + 6451190396328 z^6 + 2332850300946 z^7 + 234529149195 z^8 + 3584655360 z^9) EllipticK[(2 (-1 + Sqrt[1 - z]) (1 - z)^(1/4) + z)/(2 z)])/ (155607276825 Sqrt[2] Pi Sqrt[1 + Sqrt[1 - z]] z^3)

Standard Form

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MathML Form

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2 </mn> <mo> ⁢ </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mi> 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> 16928279775 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 531937693782 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 3375456537024 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 6538595210184 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4189539125600 </mn> <mo> ⁢ 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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]