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 Hypergeometric2F1

 http://functions.wolfram.com/07.23.03.az5l.01

 Input Form

 Hypergeometric2F1[1, 2, 13/5, z] == -(8/(5 z)) + (12 Sqrt[10 - 2 Sqrt[5]] (-1 + z)^4 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))])/ (25 (1 - z)^(22/5) z^(8/5)) - (12 Sqrt[2 (5 + Sqrt[5])] (-1 + z)^4 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))])/ (25 (1 - z)^(22/5) z^(8/5)) + (24 (-1 + z)^4 Log[1 + z^(1/5)/(1 - z)^(1/5)])/ (25 (1 - z)^(22/5) z^(8/5)) - (6 (1 + Sqrt[5]) (-1 + z)^4 Log[1 - ((1 - Sqrt[5]) z^(1/5))/(2 (1 - z)^(1/5)) + z^(2/5)/(1 - z)^(2/5)])/(25 (1 - z)^(22/5) z^(8/5)) + (6 (-1 + Sqrt[5]) (-1 + z)^4 Log[1 - ((1 + Sqrt[5]) z^(1/5))/(2 (1 - z)^(1/5)) + z^(2/5)/(1 - z)^(2/5)])/(25 (1 - z)^(22/5) z^(8/5))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", "2", ",", FractionBox["13", "5"], ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["8", RowBox[List["5", " ", "z"]]]]], "+", FractionBox[RowBox[List["12", " ", SqrtBox[RowBox[List["10", "-", RowBox[List["2", " ", SqrtBox["5"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "4"], " ", 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["25", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["22", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["8", "/", "5"]]]]]], "-", FractionBox[RowBox[List["12", " ", SqrtBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["5", "+", SqrtBox["5"]]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "4"], " ", 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["25", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["22", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["8", "/", "5"]]]]]], "+", FractionBox[RowBox[List["24", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "4"], " ", 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["25", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["22", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["8", "/", "5"]]]]]], "-", FractionBox[RowBox[List["6", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "4"], " ", 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["25", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["22", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["8", "/", "5"]]]]]], "+", FractionBox[RowBox[List["6", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["5"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "4"], " ", 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["25", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["22", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["8", "/", "5"]]]]]]]]]]]]

 MathML Form

 2 F 1 ( 1 , 2 ; 13 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["2", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[FractionBox["13", "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] 12 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 ) 4 25 ( 1 - z ) 22 / 5 z 8 / 5 - 12 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 ) 4 25 ( 1 - z ) 22 / 5 z 8 / 5 + 24 log ( z 5 1 - z 5 + 1 ) ( z - 1 ) 4 25 ( 1 - z ) 22 / 5 z 8 / 5 - 6 ( 1 + 5 ) log ( z 2 / 5 ( 1 - z ) 2 / 5 - ( 1 - 5 ) z 5 2 1 - z 5 + 1 ) ( z - 1 ) 4 25 ( 1 - z ) 22 / 5 z 8 / 5 + 6 ( - 1 + 5 ) log ( z 2 / 5 ( 1 - z ) 2 / 5 - ( 1 + 5 ) z 5 2 1 - z 5 + 1 ) ( z - 1 ) 4 25 ( 1 - z ) 22 / 5 z 8 / 5 - 8 5 z HypergeometricPFQ 1 2 13 5 z 12 10 -1 2 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 4 25 1 -1 z 22 5 z 8 5 -1 -1 12 2 5 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 4 25 1 -1 z 22 5 z 8 5 -1 24 z 1 5 1 -1 z 1 5 -1 1 z -1 4 25 1 -1 z 22 5 z 8 5 -1 -1 6 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 4 25 1 -1 z 22 5 z 8 5 -1 6 -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 4 25 1 -1 z 22 5 z 8 5 -1 -1 8 5 z -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List["1", ",", "2", ",", FractionBox["13", "5"], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox["8", RowBox[List["5", " ", "z"]]]]], "+", FractionBox[RowBox[List["12", " ", SqrtBox[RowBox[List["10", "-", RowBox[List["2", " ", SqrtBox["5"]]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "4"], " ", 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["25", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["22", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["8", "/", "5"]]]]]], "-", FractionBox[RowBox[List["12", " ", SqrtBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List["5", "+", SqrtBox["5"]]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "4"], " ", 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["25", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["22", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["8", "/", "5"]]]]]], "+", FractionBox[RowBox[List["24", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "4"], " ", 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["25", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["22", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["8", "/", "5"]]]]]], "-", FractionBox[RowBox[List["6", " ", RowBox[List["(", RowBox[List["1", "+", SqrtBox["5"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "4"], " ", 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["25", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["22", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["8", "/", "5"]]]]]], "+", FractionBox[RowBox[List["6", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SqrtBox["5"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "4"], " ", 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["25", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["22", "/", "5"]]], " ", SuperscriptBox["z", RowBox[List["8", "/", "5"]]]]]]]]]]]]

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

 2007-05-02