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

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=-15/4, b>=a

For fixed z and a=-15/4, b=5/4

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

Input Form

Hypergeometric2F1[-(15/4), 5/4, -(11/2), z] == (1/(10560 Pi^(3/2))) (((1/(-1 + z)^3) (2 Sqrt[z] (-5280 + 9360 z - 2394 z^2 - 921 z^3 - 573 z^4 - 480 z^5 + 2048 z^6) EllipticE[(1/2) (1 - Sqrt[z])]) - (1/(-1 + z)^3) (2 Sqrt[z] (-5280 + 9360 z - 2394 z^2 - 921 z^3 - 573 z^4 - 480 z^5 + 2048 z^6) EllipticE[(1/2) (1 + Sqrt[z])]) + (1/((-1 + Sqrt[z])^3 (1 + Sqrt[z])^2)) ((-10560 + 15840 Sqrt[z] + 10800 z - 20160 z^(3/2) + 2520 z^2 - 126 z^(5/2) + 363 z^3 + 558 z^(7/2) - 459 z^4 + 1032 z^(9/2) - 1056 z^5 + 1536 z^(11/2) - 2048 z^6) EllipticK[(1/2) (1 - Sqrt[z])]) + (1/((-1 + Sqrt[z])^2 (1 + Sqrt[z])^3)) ((10560 + 15840 Sqrt[z] - 10800 z - 20160 z^(3/2) - 2520 z^2 - 126 z^(5/2) - 363 z^3 + 558 z^(7/2) + 459 z^4 + 1032 z^(9/2) + 1056 z^5 + 1536 z^(11/2) + 2048 z^6) EllipticK[(1/2) (1 + Sqrt[z])])) Gamma[3/4]^2)

Standard Form

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MathML Form

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</apply> </apply> <apply> <times /> <cn type='integer'> 9360 </cn> <ci> z </ci> </apply> <cn type='integer'> -5280 </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> <power /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </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'> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> 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type='rational'> 7 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 363 </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'> 126 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 5 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2520 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 20160 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 10800 </cn> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 15840 </cn> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -10560 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <times /> <cn 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type='rational'> 3 <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]