Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9607)

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=-23/4, b>=a

For fixed z and a=-23/4, b=5/4

http://functions.wolfram.com/07.23.03.9607.01

Input Form

Hypergeometric2F1[-(23/4), 5/4, 5, -z] == (4096 Sqrt[2] (Sqrt[1 + z] (55936 + 468464 z + 1656667 z^2 + 2826516 z^3 + 6140554 z^4 + 6962900 z^5 + 4972275 z^6 + 2228864 z^7 + 575488 z^8 + 65536 z^9) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] + (55936 + 524400 z + 2125131 z^2 + 4483183 z^3 + 8967070 z^4 + 13103454 z^5 + 11935175 z^6 + 7201139 z^7 + 2804352 z^8 + 641024 z^9 + 65536 z^10) EllipticE[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - 2 (27968 + 255208 z + 1002041 z^2 + 2018940 z^3 + 950390 z^4 + 1014412 z^5 + 691257 z^6 + 297784 z^7 + 74240 z^8 + 8192 z^9) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])] - Sqrt[1 + z] (55936 + 468464 z + 1656667 z^2 + 2826516 z^3 + 6140554 z^4 + 6962900 z^5 + 4972275 z^6 + 2228864 z^7 + 575488 z^8 + 65536 z^9) EllipticK[(-1 + Sqrt[1 + z])/(1 + Sqrt[1 + z])]))/ (12814716915 Pi z^4 Sqrt[1 + Sqrt[1 + z]])

Standard Form

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MathML Form

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type='integer'> 575488 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 8 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2228864 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 7 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4972275 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6962900 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 6140554 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2826516 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1656667 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 468464 </cn> <ci> z </ci> </apply> <cn type='integer'> 55936 </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'> 12814716915 </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]