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 Hypergeometric2F1

 http://functions.wolfram.com/07.23.03.az1z.01

 Input Form

 Hypergeometric2F1[1, 1, 21/5, z] == (8 (66 - 143 z + 83 z^2))/(33 z^3) + (8 Sqrt[2 (5 + Sqrt[5])] (-1 + z)^8 ArcTan[1 - ((1 - Sqrt[5]) z^(1/5))/(4 (1 - z)^(1/5)), -((Sqrt[5/8 + Sqrt[5]/8] z^(1/5))/(1 - z)^(1/5))])/ (5 (1 - z)^(29/5) z^(16/5)) + (8 Sqrt[10 - 2 Sqrt[5]] (-1 + z)^8 ArcTan[1 - ((1 + Sqrt[5]) z^(1/5))/(4 (1 - z)^(1/5)), -((Sqrt[5/8 - Sqrt[5]/8] z^(1/5))/(1 - z)^(1/5))])/ (5 (1 - z)^(29/5) z^(16/5)) - (16 (-1 + z)^8 Log[1 + z^(1/5)/(1 - z)^(1/5)])/ (5 (1 - z)^(29/5) z^(16/5)) - (4 (-1 + Sqrt[5]) (-1 + z)^8 Log[1 - ((1 - Sqrt[5]) z^(1/5))/(2 (1 - z)^(1/5)) + z^(2/5)/(1 - z)^(2/5)])/(5 (1 - z)^(29/5) z^(16/5)) + (4 (1 + Sqrt[5]) (-1 + z)^8 Log[1 - ((1 + Sqrt[5]) z^(1/5))/(2 (1 - z)^(1/5)) + z^(2/5)/(1 - z)^(2/5)])/(5 (1 - z)^(29/5) z^(16/5))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", "1", ",", FractionBox["21", "5"], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["8", " ", RowBox[List["(", RowBox[List["66", "-", RowBox[List["143", " ", "z"]], "+", RowBox[List["83", " ", SuperscriptBox["z", "2"]]]]], ")"]]]], RowBox[List["33", " ", SuperscriptBox["z", "3"]]]], "+", FractionBox[RowBox[List["8", " ", SqrtBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["5", "+", SqrtBox["5"]]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "8"], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", SqrtBox["5"]]], ")"]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "5"]]]]]]]], ",", RowBox[List["-", FractionBox[RowBox[List[SqrtBox[RowBox[List[FractionBox["5", "8"], "+", FractionBox[SqrtBox["5"], "8"]]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "5"]]]]]]]], "]"]]]], RowBox[List["5", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["29", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["16", "/", "5"]]]]]], "+", FractionBox[RowBox[List["8", " ", SqrtBox[RowBox[List["10", "-", RowBox[List["2", " ", SqrtBox["5"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "8"], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "5"]]]]]]]], ",", RowBox[List["-", FractionBox[RowBox[List[SqrtBox[RowBox[List[FractionBox["5", "8"], "-", FractionBox[SqrtBox["5"], "8"]]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "5"]]]]]]]], "]"]]]], RowBox[List["5", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["29", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["16", "/", "5"]]]]]], "-", FractionBox[RowBox[List["16", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "8"], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[SuperscriptBox["z", RowBox[List["1", "/", "5"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "5"]]]]]], "]"]]]], RowBox[List["5", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["29", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["16", "/", "5"]]]]]], "-", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["5"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "8"], " ", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", SqrtBox["5"]]], ")"]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "5"]]]]]], "+", FractionBox[SuperscriptBox["z", RowBox[List["2", "/", "5"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["2", "/", "5"]]]]]], "]"]]]], RowBox[List["5", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["29", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["16", "/", "5"]]]]]], "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "8"], " ", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "5"]]]]]], "+", FractionBox[SuperscriptBox["z", RowBox[List["2", "/", "5"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["2", "/", "5"]]]]]], "]"]]]], RowBox[List["5", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["29", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["16", "/", "5"]]]]]]]]]]]]

 MathML Form

 2 F 1 ( 1 , 1 ; 21 5 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["1", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["1", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[FractionBox["21", "5"], 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] 8 2 ( 5 + 5 ) tan - 1 ( 1 - ( 1 - 5 ) z 5 4 1 - z 5 , - 5 8 + 5 8 z 5 1 - z 5 ) ( z - 1 ) 8 5 ( 1 - z ) 29 / 5 z 16 / 5 + 8 10 - 2 5 tan - 1 ( 1 - ( 1 + 5 ) z 5 4 1 - z 5 , - 5 8 - 5 8 z 5 1 - z 5 ) ( z - 1 ) 8 5 ( 1 - z ) 29 / 5 z 16 / 5 - 16 log ( z 5 1 - z 5 + 1 ) ( z - 1 ) 8 5 ( 1 - z ) 29 / 5 z 16 / 5 - 4 ( - 1 + 5 ) log ( z 2 / 5 ( 1 - z ) 2 / 5 - ( 1 - 5 ) z 5 2 1 - z 5 + 1 ) ( z - 1 ) 8 5 ( 1 - z ) 29 / 5 z 16 / 5 + 4 ( 1 + 5 ) log ( z 2 / 5 ( 1 - z ) 2 / 5 - ( 1 + 5 ) z 5 2 1 - z 5 + 1 ) ( z - 1 ) 8 5 ( 1 - z ) 29 / 5 z 16 / 5 + 8 ( 83 z 2 - 143 z + 66 ) 33 z 3 HypergeometricPFQ 1 1 21 5 z 8 2 5 5 1 2 1 2 1 -1 1 -1 5 1 2 z 1 5 4 1 -1 z 1 5 -1 -1 5 8 5 1 2 8 -1 1 2 z 1 5 1 -1 z 1 5 -1 z -1 8 5 1 -1 z 29 5 z 16 5 -1 8 10 -1 2 5 1 2 1 2 1 -1 1 5 1 2 z 1 5 4 1 -1 z 1 5 -1 -1 5 8 -1 5 1 2 8 -1 1 2 z 1 5 1 -1 z 1 5 -1 z -1 8 5 1 -1 z 29 5 z 16 5 -1 -1 16 z 1 5 1 -1 z 1 5 -1 1 z -1 8 5 1 -1 z 29 5 z 16 5 -1 -1 4 -1 5 1 2 z 2 5 1 -1 z 2 5 -1 -1 1 -1 5 1 2 z 1 5 2 1 -1 z 1 5 -1 1 z -1 8 5 1 -1 z 29 5 z 16 5 -1 4 1 5 1 2 z 2 5 1 -1 z 2 5 -1 -1 1 5 1 2 z 1 5 2 1 -1 z 1 5 -1 1 z -1 8 5 1 -1 z 29 5 z 16 5 -1 8 83 z 2 -1 143 z 66 33 z 3 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", "1", ",", FractionBox["21", "5"], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["8", " ", RowBox[List["(", RowBox[List["66", "-", RowBox[List["143", " ", "z"]], "+", RowBox[List["83", " ", SuperscriptBox["z", "2"]]]]], ")"]]]], RowBox[List["33", " ", SuperscriptBox["z", "3"]]]], "+", FractionBox[RowBox[List["8", " ", SqrtBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["5", "+", SqrtBox["5"]]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "8"], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", SqrtBox["5"]]], ")"]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "5"]]]]]]]], ",", RowBox[List["-", FractionBox[RowBox[List[SqrtBox[RowBox[List[FractionBox["5", "8"], "+", FractionBox[SqrtBox["5"], "8"]]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "5"]]]]]]]], "]"]]]], RowBox[List["5", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["29", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["16", "/", "5"]]]]]], "+", FractionBox[RowBox[List["8", " ", SqrtBox[RowBox[List["10", "-", RowBox[List["2", " ", SqrtBox["5"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "8"], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], RowBox[List["4", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "5"]]]]]]]], ",", RowBox[List["-", FractionBox[RowBox[List[SqrtBox[RowBox[List[FractionBox["5", "8"], "-", FractionBox[SqrtBox["5"], "8"]]]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "5"]]]]]]]], "]"]]]], RowBox[List["5", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["29", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["16", "/", "5"]]]]]], "-", FractionBox[RowBox[List["16", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "8"], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[SuperscriptBox["z", RowBox[List["1", "/", "5"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "5"]]]]]], "]"]]]], RowBox[List["5", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["29", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["16", "/", "5"]]]]]], "-", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["5"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "8"], " ", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "-", SqrtBox["5"]]], ")"]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "5"]]]]]], "+", FractionBox[SuperscriptBox["z", RowBox[List["2", "/", "5"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["2", "/", "5"]]]]]], "]"]]]], RowBox[List["5", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["29", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["16", "/", "5"]]]]]], "+", FractionBox[RowBox[List["4", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "8"], " ", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], " ", SuperscriptBox["z", RowBox[List["1", "/", "5"]]]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "5"]]]]]], "+", FractionBox[SuperscriptBox["z", RowBox[List["2", "/", "5"]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["2", "/", "5"]]]]]], "]"]]]], RowBox[List["5", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["29", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["16", "/", "5"]]]]]]]]]]]]

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

 2007-05-02