Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a85i)

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=-21/4

http://functions.wolfram.com/07.23.03.a85i.01

Input Form

Hypergeometric2F1[-(21/4), -(21/4), 6, -z] == (16384 Sqrt[2] ((-(28514304 + 747163872 z + 10634169273 z^2 + 120504827352 z^3 + 1584723786828 z^4 - 120401581743928 z^5 + 467117413896358 z^6 - 525396327335128 z^7 + 199922927611628 z^8 - 23340990842408 z^9 + 529774888169 z^10)) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - Sqrt[1 + z] (28514304 + 747163872 z + 10634169273 z^2 + 120504827352 z^3 + 1584723786828 z^4 - 120401581743928 z^5 + 467117413896358 z^6 - 525396327335128 z^7 + 199922927611628 z^8 - 23340990842408 z^9 + 529774888169 z^10) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + Sqrt[1 + z] (28514304 + 740035296 z + 10452501969 z^2 + 117976336296 z^3 + 1556401880124 z^4 - 71026172170984 z^5 + 247174276383494 z^6 - 253094730705384 z^7 + 87174280558844 z^8 - 9005507556664 z^9 + 168441498225 z^10) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + (28514304 + 747163872 z + 10634169273 z^2 + 120504827352 z^3 + 1584723786828 z^4 - 120401581743928 z^5 + 467117413896358 z^6 - 525396327335128 z^7 + 199922927611628 z^8 - 23340990842408 z^9 + 529774888169 z^10) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))/ (407563575727683375 Pi z^5 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 120401581743928 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1584723786828 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 120504827352 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 10634169273 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 747163872 </cn> <ci> z </ci> </apply> <cn type='integer'> 28514304 </cn> </apply> <apply> <ci> EllipticE </ci> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> 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<apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 247174276383494 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 71026172170984 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 1556401880124 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 117976336296 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 10452501969 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 740035296 </cn> <ci> z </ci> </apply> <cn type='integer'> 28514304 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 529774888169 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 10 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 23340990842408 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 9 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 199922927611628 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> 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</annotation-xml> </semantics> </math>

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]