Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.9453)

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=1/4

http://functions.wolfram.com/07.23.03.9453.01

Input Form

Hypergeometric2F1[-(23/4), 1/4, -(11/2), z] == (1/(221760 Pi^(3/2))) ((4 Sqrt[z] (55440 + 35280 z + 27727 z^2 + 23766 z^3 + 21504 z^4 + 20480 z^5) EllipticE[(1/2) (1 - Sqrt[z])] - 4 Sqrt[z] (55440 + 35280 z + 27727 z^2 + 23766 z^3 + 21504 z^4 + 20480 z^5) EllipticE[(1/2) (1 + Sqrt[z])] - (-221760 + 110880 Sqrt[z] + 25200 z + 70560 z^(3/2) + 19880 z^2 + 55454 z^(5/2) + 16467 z^3 + 47532 z^(7/2) + 13632 z^4 + 43008 z^(9/2) + 10240 z^5 + 40960 z^(11/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (221760 + 110880 Sqrt[z] - 25200 z + 70560 z^(3/2) - 19880 z^2 + 55454 z^(5/2) - 16467 z^3 + 47532 z^(7/2) - 13632 z^4 + 43008 z^(9/2) - 10240 z^5 + 40960 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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<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["1", "4"], ",", RowBox[List["-", FractionBox["11", "2"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["55440", "+", RowBox[List["35280", " ", "z"]], "+", RowBox[List["27727", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["23766", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["21504", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["20480", " ", SuperscriptBox["z", "5"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], "-", RowBox[List["4", " ", SqrtBox["z"], " ", RowBox[List["(", RowBox[List["55440", "+", RowBox[List["35280", " ", "z"]], "+", RowBox[List["27727", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["23766", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["21504", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["20480", " ", SuperscriptBox["z", "5"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["z"]]], ")"]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "221760"]], "+", RowBox[List["110880", " ", SqrtBox["z"]]], "+", RowBox[List["25200", " ", "z"]], "+", RowBox[List["70560", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "+", RowBox[List["19880", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["55454", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["16467", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["47532", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "+", RowBox[List["13632", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["43008", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]], "+", RowBox[List["10240", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["40960", " ", 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["221760", "+", RowBox[List["110880", " ", SqrtBox["z"]]], "-", RowBox[List["25200", " ", "z"]], "+", RowBox[List["70560", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["19880", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["55454", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "-", RowBox[List["16467", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["47532", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "-", RowBox[List["13632", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["43008", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]], "-", RowBox[List["10240", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["40960", " ", 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["221760", " ", 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]