Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a8z4)

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=-21/4, b>=a

For fixed z and a=-21/4, b=23/4

http://functions.wolfram.com/07.23.03.a8z4.01

Input Form

Hypergeometric2F1[-(21/4), 23/4, 11/2, z] == (1/(127536944535 Pi^(3/2) z^(9/2))) (8 (-2 Sqrt[z] (185640 + 716040 z + 2830347 z^2 + 15380274 z^3 - 3775121805 z^4 + 18989836320 z^5 - 38533806080 z^6 + 39129710592 z^7 - 19855835136 z^8 + 4026531840 z^9) EllipticE[(1/2) (1 - Sqrt[z])] - 2 Sqrt[z] (185640 + 716040 z + 2830347 z^2 + 15380274 z^3 - 3775121805 z^4 + 18989836320 z^5 - 38533806080 z^6 + 39129710592 z^7 - 19855835136 z^8 + 4026531840 z^9) EllipticE[(1/2) (1 + Sqrt[z])] + (-371280 + 185640 Sqrt[z] - 1153620 z + 716040 z^(3/2) - 4544865 z^2 + 2830347 z^(5/2) - 26333034 z^3 + 15380274 z^(7/2) - 397014345 z^4 - 3775121805 z^(9/2) + 3100875960 z^5 + 18989836320 z^(11/2) - 7536488960 z^6 - 38533806080 z^(13/2) + 8539668480 z^7 + 39129710592 z^(15/2) - 4680843264 z^8 - 19855835136 z^(17/2) + 1006632960 z^9 + 4026531840 z^(19/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (371280 + 185640 Sqrt[z] + 1153620 z + 716040 z^(3/2) + 4544865 z^2 + 2830347 z^(5/2) + 26333034 z^3 + 15380274 z^(7/2) + 397014345 z^4 - 3775121805 z^(9/2) - 3100875960 z^5 + 18989836320 z^(11/2) + 7536488960 z^6 - 38533806080 z^(13/2) - 8539668480 z^7 + 39129710592 z^(15/2) + 4680843264 z^8 - 19855835136 z^(17/2) - 1006632960 z^9 + 4026531840 z^(19/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[1/4]^2)

Standard Form

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MathML Form

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<times /> <cn type='integer'> -2 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 4026531840 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 19855835136 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 39129710592 </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'> 38533806080 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 18989836320 </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'> 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type='integer'> 4026531840 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 19855835136 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 39129710592 </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'> 38533806080 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 18989836320 </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'> 3775121805 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 15380274 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2830347 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 716040 </cn> <ci> z </ci> </apply> <cn type='integer'> 185640 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4026531840 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 19 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1006632960 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 19855835136 </cn> <apply> 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type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 38533806080 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 13 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 7536488960 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 18989836320 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 11 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3100875960 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3775121805 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 9 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 397014345 </cn> <apply> <power /> <ci> z </ci> 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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]