Gauss hypergeometric function 2F1: Specific values (formula 07.23.03.a84y)

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=-21/4, b>=a

For fixed z and a=-21/4, b=-21/4

http://functions.wolfram.com/07.23.03.a84y.01

Input Form

Hypergeometric2F1[-(21/4), -(21/4), -(5/2), z] == (1/(3360 Pi^(3/2))) ((-8 (-210 + 2289 z - 13664 z^2 + 96558 z^3 + 208410 z^4 + 34297 z^5) EllipticE[(1/2) (1 - Sqrt[z])] - 8 (-210 + 2289 z - 13664 z^2 + 96558 z^3 + 208410 z^4 + 34297 z^5) EllipticE[(1/2) (1 + Sqrt[z])] + (-840 - 420 Sqrt[z] + 9156 z + 4543 z^(3/2) - 54656 z^2 - 26964 z^(5/2) + 386232 z^3 + 603114 z^(7/2) + 833640 z^4 + 660832 z^(9/2) + 137188 z^5 + 69615 z^(11/2)) EllipticK[(1/2) (1 - Sqrt[z])] + (-840 + 420 Sqrt[z] + 9156 z - 4543 z^(3/2) - 54656 z^2 + 26964 z^(5/2) + 386232 z^3 - 603114 z^(7/2) + 833640 z^4 - 660832 z^(9/2) + 137188 z^5 - 69615 z^(11/2)) EllipticK[(1/2) (1 + Sqrt[z])]) Gamma[1/4]^2)

Standard Form

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MathML Form

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<cn type='rational'> 1 <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["21", "4"]]], ",", RowBox[List["-", FractionBox["21", "4"]]], ",", RowBox[List["-", FractionBox["5", "2"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "8"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "210"]], "+", RowBox[List["2289", " ", "z"]], "-", RowBox[List["13664", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["96558", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["208410", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["34297", " ", SuperscriptBox["z", "5"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SqrtBox["z"]]], ")"]]]], "]"]]]], "-", RowBox[List["8", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "210"]], "+", RowBox[List["2289", " ", "z"]], "-", RowBox[List["13664", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["96558", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["208410", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["34297", " ", 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["-", "840"]], "-", RowBox[List["420", " ", SqrtBox["z"]]], "+", RowBox[List["9156", " ", "z"]], "+", RowBox[List["4543", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["54656", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["26964", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["386232", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["603114", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "+", RowBox[List["833640", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["660832", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]], "+", RowBox[List["137188", " ", SuperscriptBox["z", "5"]]], "+", RowBox[List["69615", " ", 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[RowBox[List["-", "840"]], "+", RowBox[List["420", " ", SqrtBox["z"]]], "+", RowBox[List["9156", " ", "z"]], "-", RowBox[List["4543", " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]]]], "-", RowBox[List["54656", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["26964", " ", SuperscriptBox["z", RowBox[List["5", "/", "2"]]]]], "+", RowBox[List["386232", " ", SuperscriptBox["z", "3"]]], "-", RowBox[List["603114", " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]]]], "+", RowBox[List["833640", " ", SuperscriptBox["z", "4"]]], "-", RowBox[List["660832", " ", SuperscriptBox["z", RowBox[List["9", "/", "2"]]]]], "+", RowBox[List["137188", " ", SuperscriptBox["z", "5"]]], "-", RowBox[List["69615", " ", 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["1", "4"], "]"]], "2"]]], RowBox[List["3360", " ", 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]