Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a9rf)

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=17/4

http://functions.wolfram.com/07.23.03.a9rf.01

Input Form

Hypergeometric2F1[-(19/4), 17/4, 5, -z] == (4096 Sqrt[2] (Sqrt[1 + z] (-80256 + 103664 z - 209209 z^2 + 851466 z^3 + 20319715 z^4 + 58141312 z^5 + 69568512 z^6 + 38862848 z^7 + 8388608 z^8) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + (-80256 + 23408 z - 105545 z^2 + 642257 z^3 + 21171181 z^4 + 78461027 z^5 + 127709824 z^6 + 108431360 z^7 + 47251456 z^8 + 8388608 z^9) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - 2 (-40128 + 21736 z - 62909 z^2 + 342342 z^3 + 3784835 z^4 + 9168776 z^5 + 9927168 z^6 + 5152768 z^7 + 1048576 z^8) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - Sqrt[1 + z] (-80256 + 103664 z - 209209 z^2 + 851466 z^3 + 20319715 z^4 + 58141312 z^5 + 69568512 z^6 + 38862848 z^7 + 8388608 z^8) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))/ (27460107675 Pi z^4 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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<apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 342342 </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'> 62909 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 21736 </cn> <ci> z </ci> </apply> <cn type='integer'> -40128 </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> 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</apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 209209 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 103664 </cn> <ci> z </ci> </apply> <cn type='integer'> -80256 </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> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 27460107675 </cn> <pi /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </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='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </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]