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 Hypergeometric2F1

 http://functions.wolfram.com/07.23.03.c57m.01

 Input Form

 Hypergeometric2F1[15/8, 21/8, 6, z] == (1/(72747675 Pi z^5)) (524288 2^(1/4) (1 + Sqrt[1 - z])^(1/4) (2 Sqrt[2] (-98304 + 177792 z - 82057 z^2 + 4389 z^3) EllipticE[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] + (12288 (8 Sqrt[2] + 5 Sqrt[1 + Sqrt[1 - z]] + 3 Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z]) - 48 (3704 Sqrt[2] + 2105 Sqrt[1 + Sqrt[1 - z]] + 831 Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z]) z + (82057 Sqrt[2] + 37780 Sqrt[1 + Sqrt[1 - z]] + 4389 Sqrt[1 + Sqrt[1 - z]] Sqrt[1 - z]) z^2 - 4389 Sqrt[2] z^3) EllipticK[1/2 - 1/(Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["15", "8"], ",", FractionBox["21", "8"], ",", "6", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["72747675", " ", "\[Pi]", " ", SuperscriptBox["z", "5"]]]], RowBox[List["(", RowBox[List["524288", " ", SuperscriptBox["2", RowBox[List["1", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "98304"]], "+", RowBox[List["177792", " ", "z"]], "-", RowBox[List["82057", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["4389", " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", 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[RowBox[List["(", RowBox[List[RowBox[List["12288", " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", SqrtBox["2"]]], "+", RowBox[List["5", " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]], "+", RowBox[List["3", " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]]], ")"]]]], "-", RowBox[List["48", " ", RowBox[List["(", RowBox[List[RowBox[List["3704", " ", SqrtBox["2"]]], "+", RowBox[List["2105", " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]], "+", RowBox[List["831", " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]]], ")"]], " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["82057", " ", SqrtBox["2"]]], "+", RowBox[List["37780", " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]], "+", RowBox[List["4389", " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]]], ")"]], " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["4389", " ", SqrtBox["2"], " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox["1", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]]]], ")"]]]], ")"]]]]]]]]

 MathML Form

 2 F 1 ( 15 8 , 21 8 ; 6 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["15", "8"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["21", "8"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox["6", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox["z", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]], HypergeometricPFQ] 1 72747675 π z 5 ( 524288 2 4 1 - z + 1 4 ( 2 2 ( 4389 z 3 - 82057 z 2 + 177792 z - 98304 ) E ( 1 2 - 1 2 1 - z + 1 ) + ( - 4389 2 z 3 + ( 4389 1 - z 1 - z + 1 + 37780 1 - z + 1 + 82057 2 ) z 2 - 48 ( 831 1 - z 1 - z + 1 + 2105 1 - z + 1 + 3704 2 ) z + 12288 ( 3 1 - z 1 - z + 1 + 5 1 - z + 1 + 8 2 ) ) K ( 1 2 - 1 2 1 - z + 1 ) ) ) HypergeometricPFQ 15 8 21 8 6 z 1 72747675 z 5 -1 524288 2 1 4 1 -1 z 1 2 1 1 4 2 2 1 2 4389 z 3 -1 82057 z 2 177792 z -98304 EllipticE 1 2 -1 1 2 1 2 1 -1 z 1 2 1 1 2 -1 -4389 2 1 2 z 3 4389 1 -1 z 1 2 1 -1 z 1 2 1 1 2 37780 1 -1 z 1 2 1 1 2 82057 2 1 2 z 2 -1 48 831 1 -1 z 1 2 1 -1 z 1 2 1 1 2 2105 1 -1 z 1 2 1 1 2 3704 2 1 2 z 12288 3 1 -1 z 1 2 1 -1 z 1 2 1 1 2 5 1 -1 z 1 2 1 1 2 8 2 1 2 EllipticK 1 2 -1 1 2 1 2 1 -1 z 1 2 1 1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["15", "8"], ",", FractionBox["21", "8"], ",", "6", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["524288", " ", SuperscriptBox["2", RowBox[List["1", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "98304"]], "+", RowBox[List["177792", " ", "z"]], "-", RowBox[List["82057", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["4389", " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", 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[RowBox[List["(", RowBox[List[RowBox[List["12288", " ", RowBox[List["(", RowBox[List[RowBox[List["8", " ", SqrtBox["2"]]], "+", RowBox[List["5", " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]], "+", RowBox[List["3", " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]]], ")"]]]], "-", RowBox[List["48", " ", RowBox[List["(", RowBox[List[RowBox[List["3704", " ", SqrtBox["2"]]], "+", RowBox[List["2105", " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]], "+", RowBox[List["831", " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]]], ")"]], " ", "z"]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["82057", " ", SqrtBox["2"]]], "+", RowBox[List["37780", " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]], "+", RowBox[List["4389", " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]]], ")"]], " ", SuperscriptBox["z", "2"]]], "-", RowBox[List["4389", " ", SqrtBox["2"], " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", 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["72747675", " ", "\[Pi]", " ", SuperscriptBox["z", "5"]]]]]]]]

 Date Added to functions.wolfram.com (modification date)

 2007-05-02