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 Hypergeometric2F1

 http://functions.wolfram.com/07.23.03.c4fh.01

 Input Form

 Hypergeometric2F1[13/8, 25/8, 3, z] == (256 2^(1/4) (4 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) (4 - 5 z + 10 z^2) EllipticE[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])] - (8 (1 + Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) + Sqrt[1 - z]) - (13 + 10 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) + 10 Sqrt[1 - z]) z + 5 (1 + 4 Sqrt[2] Sqrt[1 + Sqrt[1 - z]] (1 - z)^(1/4) + 4 Sqrt[1 - z]) z^2) EllipticK[1/2 - (1 - z)^(1/4)/ (Sqrt[2] Sqrt[1 + Sqrt[1 - z]])]))/(2295 Pi (1 + Sqrt[1 - z])^(1/4) (-1 + z)^2 z^2)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["13", "8"], ",", FractionBox["25", "8"], ",", "3", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["256", " ", SuperscriptBox["2", RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List["4", "-", RowBox[List["5", " ", "z"]], "+", RowBox[List["10", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["8", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]]], "+", SqrtBox[RowBox[List["1", "-", "z"]]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["13", "+", RowBox[List["10", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]]], "+", RowBox[List["10", " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]]], ")"]], " ", "z"]], "+", RowBox[List["5", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]]], "+", RowBox[List["4", " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]]], ")"]], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]]]], ")"]]]], ")"]], "/", RowBox[List["(", RowBox[List["2295", " ", "\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]], ")"]]]]]]]]

 MathML Form

 2 F 1 ( 13 8 , 25 8 ; 3 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox[FractionBox["13", "8"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox[FractionBox["25", "8"], HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox["3", 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] ( 256 2 4 ( 4 2 1 - z + 1 1 - z 4 ( 10 z 2 - 5 z + 4 ) E ( 1 2 - 1 - z 4 2 1 - z + 1 ) - ( 5 ( 4 2 1 - z 4 1 - z + 1 + 4 1 - z + 1 ) z 2 - ( 10 2 1 - z 4 1 - z + 1 + 10 1 - z + 13 ) z + 8 ( 2 1 - z 4 1 - z + 1 + 1 - z + 1 ) ) K ( 1 2 - 1 - z 4 2 1 - z + 1 ) ) ) / ( 2295 π 1 - z + 1 4 ( z - 1 ) 2 z 2 ) HypergeometricPFQ 13 8 25 8 3 z 256 2 1 4 4 2 1 2 1 -1 z 1 2 1 1 2 1 -1 z 1 4 10 z 2 -1 5 z 4 EllipticE 1 2 -1 1 -1 z 1 4 2 1 2 1 -1 z 1 2 1 1 2 -1 -1 5 4 2 1 2 1 -1 z 1 4 1 -1 z 1 2 1 1 2 4 1 -1 z 1 2 1 z 2 -1 10 2 1 2 1 -1 z 1 4 1 -1 z 1 2 1 1 2 10 1 -1 z 1 2 13 z 8 2 1 2 1 -1 z 1 4 1 -1 z 1 2 1 1 2 1 -1 z 1 2 1 EllipticK 1 2 -1 1 -1 z 1 4 2 1 2 1 -1 z 1 2 1 1 2 -1 2295 1 -1 z 1 2 1 1 4 z -1 2 z 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List[FractionBox["13", "8"], ",", FractionBox["25", "8"], ",", "3", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["256", " ", SuperscriptBox["2", RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["(", RowBox[List["4", "-", RowBox[List["5", " ", "z"]], "+", RowBox[List["10", " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["EllipticE", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["8", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]]], "+", SqrtBox[RowBox[List["1", "-", "z"]]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["13", "+", RowBox[List["10", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]]], "+", RowBox[List["10", " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]]], ")"]], " ", "z"]], "+", RowBox[List["5", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["4", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]]], "+", RowBox[List["4", " ", SqrtBox[RowBox[List["1", "-", "z"]]]]]]], ")"]], " ", SuperscriptBox["z", "2"]]]]], ")"]], " ", RowBox[List["EllipticK", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List["2295", " ", "\[Pi]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", SqrtBox[RowBox[List["1", "-", "z"]]]]], ")"]], RowBox[List["1", "/", "4"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "2"], " ", SuperscriptBox["z", "2"]]]]]]]]

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

 2007-05-02