Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b7es)

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=-43/8, b>=a

For fixed z and a=-43/8, b=-39/8

http://functions.wolfram.com/07.23.03.b7es.01

Input Form

Hypergeometric2F1[-(43/8), -(39/8), 6, z] == (524288 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (744433876992 - 18823126897536 z + 258312080595309 z^2 - 2823609888208200 z^3 + 35940886367780940 z^4 + 1316776292849828168 z^5 + 4094609299213566766 z^6 + 3680348121443577736 z^7 + 1064778161010508300 z^8 + 83452279558304120 z^9 + 792715055646605 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) (744433876992 - 18823126897536 z + 258312080595309 z^2 - 2823609888208200 z^3 + 35940886367780940 z^4 + 1316776292849828168 z^5 + 4094609299213566766 z^6 + 3680348121443577736 z^7 + 1064778161010508300 z^8 + 83452279558304120 z^9 + 792715055646605 z^10) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - Sqrt[1 - z] (744433876992 - 18823126897536 z + 258312080595309 z^2 - 2823609888208200 z^3 + 35940886367780940 z^4 + 1316776292849828168 z^5 + 4094609299213566766 z^6 + 3680348121443577736 z^7 + 1064778161010508300 z^8 + 83452279558304120 z^9 + 792715055646605 z^10) EllipticK[1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (-744433876992 + 19102289601408 z - 265294419630045 z^2 + 2918587364845464 z^3 - 36974251825814940 z^4 + 904158893798349832 z^5 + 5871828894112932002 z^6 + 8848770565426824680 z^7 + 4293322519312874756 z^8 + 643915974730236280 z^9 + 20534511640815955 z^10) EllipticK[ 1/2 - (1 - z)^(1/4)/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/ (578744081474953421602575 Pi (1 + Sqrt[1 - z])^(1/4) z^5)

Standard Form

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MathML Form

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<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> 20534511640815955 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 10 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 643915974730236280 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 9 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4293322519312874756 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 8 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 8848770565426824680 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 7 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 5871828894112932002 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mrow> <mo> + </mo> <mrow> 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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]