Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b7ux)

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=15/8

http://functions.wolfram.com/07.23.03.b7ux.01

Input Form

Hypergeometric2F1[-(43/8), 15/8, 5, z] == (1/(48511109960775 Pi z^4)) (32768 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (-4 (-27608064 + 182040672 z - 423746037 z^2 + 115500924 z^3 - 663431630 z^4 + 1062496260 z^5 - 939792725 z^6 + 491866440 z^7 - 143367120 z^8 + 18033600 z^9) EllipticE[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z] (3451008 - 20247711 z + 38500308 z^2 - 481253850 z^3 + 864966460 z^4 - 820288335 z^5 + 452793640 z^6 - 137957040 z^7 + 18033600 z^8) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 5 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (3451008 - 22188903 z + 49500396 z^2 + 288752310 z^3 - 675045860 z^4 + 798653625 z^5 - 575890640 z^6 + 256077120 z^7 - 64920960 z^8 + 7213440 z^9) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + 2 (-27608064 + 182040672 z - 423746037 z^2 + 115500924 z^3 - 663431630 z^4 + 1062496260 z^5 - 939792725 z^6 + 491866440 z^7 - 143367120 z^8 + 18033600 z^9) EllipticK[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))

Standard Form

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MathML Form

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type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 491866440 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 939792725 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1062496260 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 663431630 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 115500924 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 423746037 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> 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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]