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

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=-27/8, b>=a

For fixed z and a=-27/8, b=47/8

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

Input Form

Hypergeometric2F1[-(27/8), 47/8, 2, z] == (1/(1830673845 Pi z)) (8 2^(3/4) (1 + Sqrt[1 - z])^(1/4) (-4 (1341153 - 160078120 z + 1130653680 z^2 - 2270324160 z^3 + 1353580800 z^4) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - 170 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1341153 - 13493584 z + 39396240 z^2 - 43430400 z^3 + 15924480 z^4) EllipticK[ 1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (1/Sqrt[1 - z]) (3 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (76445721 - 817995800 z + 2609338160 z^2 - 3217830720 z^3 + 1353580800 z^4) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]) + 2 (1341153 - 160078120 z + 1130653680 z^2 - 2270324160 z^3 + 1353580800 z^4) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))

Standard Form

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MathML Form

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</apply> </apply> </apply> </apply> </apply> </apply> </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["27", "8"]]], ",", FractionBox["47", "8"], ",", "2", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["8", " ", SuperscriptBox["2", RowBox[List["3", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", RowBox[List["(", RowBox[List["1341153", "-", RowBox[List["160078120", " ", "z"]], "+", RowBox[List["1130653680", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2270324160", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["1353580800", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox["1", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "-", RowBox[List["170", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", RowBox[List["(", RowBox[List["1341153", "-", RowBox[List["13493584", " ", "z"]], "+", RowBox[List["39396240", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["43430400", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["15924480", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox["1", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "+", FractionBox[RowBox[List["3", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", RowBox[List["(", RowBox[List["76445721", "-", RowBox[List["817995800", " ", "z"]], "+", RowBox[List["2609338160", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["3217830720", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["1353580800", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox["1", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], SqrtBox[RowBox[List["1", "-", "z"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List["1341153", "-", RowBox[List["160078120", " ", "z"]], "+", RowBox[List["1130653680", " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["2270324160", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["1353580800", " ", SuperscriptBox["z", "4"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox["1", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List["1830673845", " ", "\[Pi]", " ", "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]