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

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=-31/8, b>=a

For fixed z and a=-31/8, b=43/8

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

Input Form

Hypergeometric2F1[-(31/8), 43/8, 1, z] == (2 2^(1/4) ((1/Sqrt[1 - z]) (2 (23312719 - 345621288 z + 1306597968 z^2 - 1792879296 z^3 + 812148480 z^4) EllipticE[ 2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]) + (1/Sqrt[1 - z]) (Sqrt[2 - 2 Sqrt[1 - z]] (23312719 - 345621288 z + 1306597968 z^2 - 1792879296 z^3 + 812148480 z^4) EllipticE[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])]) + 34 (142691 - 7383792 z + 45045936 z^2 - 83965440 z^3 + 47773440 z^4) EllipticK[2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])] - (1/Sqrt[1 - z]) ((23312719 - 345621288 z + 1306597968 z^2 - 1792879296 z^3 + 812148480 z^4) EllipticK[ 2 - (2 Sqrt[2])/(Sqrt[2] + Sqrt[1 - Sqrt[1 - z]])])))/ (28164213 Pi Sqrt[Sqrt[2] + Sqrt[1 - Sqrt[1 - z]]])

Standard Form

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MathML Form

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<cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List[RowBox[List["-", FractionBox["31", "8"]]], ",", FractionBox["43", "8"], ",", "1", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", SuperscriptBox["2", RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["23312719", "-", RowBox[List["345621288", " ", "z"]], "+", RowBox[List["1306597968", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["1792879296", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["812148480", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List["2", "-", FractionBox[RowBox[List["2", " ", SqrtBox["2"]]], RowBox[List[SqrtBox["2"], "+", SqrtBox[RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], SqrtBox[RowBox[List["1", "-", "z"]]]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["2", "-", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]], " ", RowBox[List["(", RowBox[List["23312719", "-", RowBox[List["345621288", " ", "z"]], "+", RowBox[List["1306597968", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["1792879296", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["812148480", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List["2", "-", FractionBox[RowBox[List["2", " ", SqrtBox["2"]]], RowBox[List[SqrtBox["2"], "+", SqrtBox[RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], SqrtBox[RowBox[List["1", "-", "z"]]]], "+", RowBox[List["34", " ", RowBox[List["(", RowBox[List["142691", "-", RowBox[List["7383792", " ", "z"]], "+", RowBox[List["45045936", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["83965440", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["47773440", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List["2", "-", FractionBox[RowBox[List["2", " ", SqrtBox["2"]]], RowBox[List[SqrtBox["2"], "+", SqrtBox[RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["23312719", "-", RowBox[List["345621288", " ", "z"]], "+", RowBox[List["1306597968", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["1792879296", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["812148480", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List["2", "-", FractionBox[RowBox[List["2", " ", SqrtBox["2"]]], RowBox[List[SqrtBox["2"], "+", SqrtBox[RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], SqrtBox[RowBox[List["1", "-", "z"]]]]]], ")"]]]], RowBox[List["28164213", " ", "\[Pi]", " ", SqrtBox[RowBox[List[SqrtBox["2"], "+", SqrtBox[RowBox[List["1", "-", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]]]]]]]

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]