Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.bb87)

Hypergeometric2F1

$Mathematica Notation$

Hypergeometric Functions

Hypergeometric2F1[a,b,c,z]

Specific values

For rational parameters with denominators 8 and fixed z and a<0

For fixed z and a=-37/8, b>=a

For fixed z and a=-37/8, b=-33/8

http://functions.wolfram.com/07.23.03.bb87.01

Input Form

Hypergeometric2F1[-(37/8), -(33/8), 5, z] == (65536 2^(1/4) (4 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-513666560 + 11958799600 z - 162050763875 z^2 + 2192021882050 z^3 + 173760625788115 z^4 + 566030966141452 z^5 + 467671668096115 z^6 + 106801208128930 z^7 + 4842348695485 z^8) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - 2 Sqrt[1 - z] (-513666560 + 11958799600 z - 162050763875 z^2 + 2192021882050 z^3 + 173760625788115 z^4 + 566030966141452 z^5 + 467671668096115 z^6 + 106801208128930 z^7 + 4842348695485 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - 2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (-513666560 + 11958799600 z - 162050763875 z^2 + 2192021882050 z^3 + 173760625788115 z^4 + 566030966141452 z^5 + 467671668096115 z^6 + 106801208128930 z^7 + 4842348695485 z^8) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (1027333120 - 24302849120 z + 332965285675 z^2 - 4503185311625 z^3 - 91721610153905 z^4 - 114829529715749 z^5 + 95990030441809 z^6 + 99906782336725 z^7 + 14633300931445 z^8 + 214521701625 z^9) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (8329219008730995375 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) z^4)

Standard Form

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MathML Form

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<msqrt> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mn> 8329219008730995375 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mroot> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <mroot> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HypergeometricPFQ </ci> <list> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 37 <sep /> 8 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn 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<cn type='integer'> 4503185311625 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 332965285675 </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'> 24302849120 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> 1027333120 </cn> </apply> <apply> <ci> EllipticK </ci> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z 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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]