Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9715)

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=9/4

http://functions.wolfram.com/07.23.03.9715.01

Input Form

Hypergeometric2F1[-(23/4), 9/4, 1/2, z] == (1/(3315 Pi^(3/2))) ((32 Sqrt[z] (-3051 + 27327 z - 80380 z^2 + 107296 z^3 - 67584 z^4 + 16384 z^5) EllipticE[(1/2) (1 - Sqrt[z])] - 32 Sqrt[z] (-3051 + 27327 z - 80380 z^2 + 107296 z^3 - 67584 z^4 + 16384 z^5) EllipticE[(1/2) (1 + Sqrt[z])] + (3315 + 48816 Sqrt[z] - 61782 z - 437232 z^(3/2) + 235363 z^2 + 1286080 z^(5/2) - 363136 z^3 - 1716736 z^(7/2) + 251904 z^4 + 1081344 z^(9/2) - 65536 z^5 - 262144 z^(11/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (3315 - 48816 Sqrt[z] - 61782 z + 437232 z^(3/2) + 235363 z^2 - 1286080 z^(5/2) - 363136 z^3 + 1716736 z^(7/2) + 251904 z^4 - 1081344 z^(9/2) - 65536 z^5 + 262144 z^(11/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[3/4]^2)

Standard Form

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MathML Form

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type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 3315 </cn> </apply> <apply> <ci> EllipticK </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> </apply> <apply> <power /> <apply> <ci> Gamma </ci> <cn type='rational'> 3 <sep /> 4 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["23", "4"]]], ",", FractionBox["9", "4"], ",", FractionBox["1", "2"], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["32", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3051"]], "+", RowBox[List["27327", " ", "z"]], "-", RowBox[List["80380", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["107296", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["67584", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["16384", " ", SuperscriptBox["z", "5"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], "-", RowBox[List["32", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "3051"]], "+", RowBox[List["27327", " ", "z"]], "-", RowBox[List["80380", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["107296", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["67584", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["16384", " ", SuperscriptBox["z", "5"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["3315", "+", RowBox[List["48816", " ", SqrtBox["z"]]], "-", RowBox[List["61782", " ", "z"]], "-", RowBox[List["437232", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["235363", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["1286080", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "-", RowBox[List["363136", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["1716736", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "+", RowBox[List["251904", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["1081344", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]], "-", RowBox[List["65536", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["262144", " ", SuperscriptBox["z", RowBox[List["11", "/", "2"]]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["3315", "-", RowBox[List["48816", " ", SqrtBox["z"]]], "-", RowBox[List["61782", " ", "z"]], "+", RowBox[List["437232", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["235363", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["1286080", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "-", RowBox[List["363136", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["1716736", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "+", RowBox[List["251904", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["1081344", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]], "-", RowBox[List["65536", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["262144", " ", SuperscriptBox["z", RowBox[List["11", "/", "2"]]]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", FractionBox["3", "4"], "]"]], "2"]]], RowBox[List["3315", " ", SuperscriptBox["\[Pi]", RowBox[List["3", "/", "2"]]]]]]]]]]

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]