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

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=19/4, b>=a

For fixed z and a=19/4, b=23/4

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

Input Form

Hypergeometric2F1[19/4, 23/4, 3/2, z] == (1/(10138590 Pi^(3/2))) ((-((1/(-1 + z)^9) (2 (2470005 + 21329580 z + 28422462 z^2 + 6453164 z^3 + 45045 z^4) EllipticE[(1/2) (1 - Sqrt[z])])) - (1/(-1 + z)^9) (2 (2470005 + 21329580 z + 28422462 z^2 + 6453164 z^3 + 45045 z^4) EllipticE[(1/2) (1 + Sqrt[z])]) + ((129285 + 2340720 Sqrt[z] + 3483900 z + 17845680 z^(3/2) + 8172270 z^2 + 20250192 z^(5/2) + 2849564 z^3 + 3603600 z^(7/2) + 45045 z^4) EllipticK[(1/2) (1 - Sqrt[z])])/((-1 + Sqrt[z])^9 (1 + Sqrt[z])^8 Sqrt[z]) + ((129285 - 2340720 Sqrt[z] + 3483900 z - 17845680 z^(3/2) + 8172270 z^2 - 20250192 z^(5/2) + 2849564 z^3 - 3603600 z^(7/2) + 45045 z^4) EllipticK[(1/2) (1 + Sqrt[z])])/ ((-1 + Sqrt[z])^8 (1 + Sqrt[z])^9 Sqrt[z])) Gamma[1/4]^2)

Standard Form

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MathML Form

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type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 8 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 9 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 45045 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3603600 </cn> <apply> <power /> <ci> z 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type='integer'> 129285 </cn> </apply> <apply> <ci> EllipticK </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> <apply> <power /> <apply> <ci> Gamma </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </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]