Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b4ml)

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=-47/8, b>=a

For fixed z and a=-47/8, b=-43/8

http://functions.wolfram.com/07.23.03.b4ml.01

Input Form

Hypergeometric2F1[-(47/8), -(43/8), 6, z] == (524288 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (34988392218624 - 975438106501248 z + 14911413601847139 z^2 - 183849351428045160 z^3 + 2680174503857001060 z^4 + 118505635210236713736 z^5 + 451625373332608468258 z^6 + 522632472770393045736 z^7 + 211876831467135670020 z^8 + 27694640097894863480 z^9 + 789620275230014355 z^10) EllipticE[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (34988392218624 - 975438106501248 z + 14911413601847139 z^2 - 183849351428045160 z^3 + 2680174503857001060 z^4 + 118505635210236713736 z^5 + 451625373332608468258 z^6 + 522632472770393045736 z^7 + 211876831467135670020 z^8 + 27694640097894863480 z^9 + 789620275230014355 z^10) EllipticK[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[1 - z] (34988392218624 - 975438106501248 z + 14911413601847139 z^2 - 183849351428045160 z^3 + 2680174503857001060 z^4 + 118505635210236713736 z^5 + 451625373332608468258 z^6 + 522632472770393045736 z^7 + 211876831467135670020 z^8 + 27694640097894863480 z^9 + 789620275230014355 z^10) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (-34988392218624 + 988558753583232 z - 15273615214848627 z^2 + 189343016973142929 z^3 - 2747640896778826620 z^4 + 74554168365233661444 z^5 + 588957074367379262198 z^6 + 1114374699039371658462 z^7 + 722646622916800383636 z^8 + 162829861285877580340 z^9 + 10396999728008303205 z^10 + 82930761618910425 z^11) EllipticK[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (50350735088320947679424025 Pi (1 + Sqrt[1 - z])^(1/4) z^5)

Standard Form

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MathML Form

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</mo> <mrow> <mi> K </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 4 </mn> </mroot> <mrow> <msqrt> <mn> 2 </mn> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 82930761618910425 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 11 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 10396999728008303205 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 10 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 162829861285877580340 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 722646622916800383636 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 1114374699039371658462 </mn> <mo> ⁢ </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]