Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.b8o9)

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=-41/8, b>=a

For fixed z and a=-41/8, b=-37/8

http://functions.wolfram.com/07.23.03.b8o9.01

Input Form

Hypergeometric2F1[-(41/8), -(37/8), 4, z] == (2048 2^(1/4) (2 Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (5265082240 - 162682814525 z + 3836723015125 z^2 + 503016418151395 z^3 + 2556977163070765 z^4 + 3375023809034017 z^5 + 1347048314473015 z^6 + 139450766366905 z^7 + 1693622228935 z^8) EllipticE[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[1 - z] (5265082240 - 162682814525 z + 3836723015125 z^2 + 503016418151395 z^3 + 2556977163070765 z^4 + 3375023809034017 z^5 + 1347048314473015 z^6 + 139450766366905 z^7 + 1693622228935 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] - Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z] (5265082240 - 162682814525 z + 3836723015125 z^2 + 503016418151395 z^3 + 2556977163070765 z^4 + 3375023809034017 z^5 + 1347048314473015 z^6 + 139450766366905 z^7 + 1693622228935 z^8) EllipticK[1/2 - Sqrt[1 - z]/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])] + (-5265082240 + 164657220365 z - 3897189193975 z^2 - 147941739893095 z^3 - 388337249816395 z^4 + 44943698954423 z^5 + 370451073002507 z^6 + 118448878040675 z^7 + 6173136767735 z^8) EllipticK[1/2 - Sqrt[1 - z]/(Sqrt[2] Sqrt[1 + Sqrt[1 - z] - z])]))/ (362139956901347625 Pi (1 + Sqrt[1 - z])^(1/4) (1 - z)^(1/4) z^3)

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 2 </mn> </msub> <msub> <mi> F </mi> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 41 </mn> <mn> 8 </mn> </mfrac> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mn> 37 </mn> <mn> 8 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mn> 4 </mn> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[RowBox[List["-", FractionBox["41", "8"]]], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[RowBox[List["-", FractionBox["37", "8"]]], HypergeometricPFQ, 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<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> 362139956901347625 </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> 3 </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'> 41 <sep /> 8 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 37 <sep /> 8 </cn> </apply> </list> <list> <cn type='integer'> 4 </cn> </list> <ci> z </ci> </apply> <apply> <times /> <apply> 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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]