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 Hypergeometric2F1

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

 Input Form

 Hypergeometric2F1[4, 5, -(1/4), z] == (1/2097152) ((1/(-1 + z)^9) (8 (-262144 + 23330816 z + 205864621 z^2 + 266720156 z^3 + 59101856 z^4 + 424320 z^5)) + (1/(1 - z)^(37/4)) (139230 Sqrt[2] z^(5/4) (663 + 3536 z + 3264 z^2 + 512 z^3) ArcTan[1 - z^(1/4)/(Sqrt[2] (1 - z)^(1/4)), -(z^(1/4)/(Sqrt[2] (1 - z)^(1/4)))]) + (1/(1 - z)^(37/4)) (139230 Sqrt[2] z^(5/4) (663 + 3536 z + 3264 z^2 + 512 z^3) ArcTan[1 + z^(1/4)/(Sqrt[2] (1 - z)^(1/4)), -(z^(1/4)/(Sqrt[2] (1 - z)^(1/4)))]) - (69615 Sqrt[2] z^(5/4) (663 + 3536 z + 3264 z^2 + 512 z^3) Log[1 - (Sqrt[2] z^(1/4))/(1 - z)^(1/4) + Sqrt[z]/Sqrt[1 - z]])/ (1 - z)^(37/4) + (69615 Sqrt[2] z^(5/4) (663 + 3536 z + 3264 z^2 + 512 z^3) Log[1 + (Sqrt[2] z^(1/4))/(1 - z)^(1/4) + Sqrt[z]/Sqrt[1 - z]])/(1 - z)^(37/4))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List["Hypergeometric2F1", "[", RowBox[List["4", ",", "5", ",", RowBox[List["-", FractionBox["1", "4"]]], ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2097152"], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "9"]], RowBox[List["8", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "262144"]], "+", RowBox[List["23330816", " ", "z"]], "+", RowBox[List["205864621", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["266720156", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["59101856", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["424320", " ", SuperscriptBox["z", "5"]]]]], ")"]]]]]], "+", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["37", "/", "4"]]]], RowBox[List["139230", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["5", "/", "4"]]], " ", RowBox[List["(", RowBox[List["663", "+", RowBox[List["3536", " ", "z"]], "+", RowBox[List["3264", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["512", " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", FractionBox[SuperscriptBox["z", RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]]]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]]]]]]]], "]"]]]]]], "+", RowBox[List[FractionBox["1", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["37", "/", "4"]]]], RowBox[List["139230", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["5", "/", "4"]]], " ", RowBox[List["(", RowBox[List["663", "+", RowBox[List["3536", " ", "z"]], "+", RowBox[List["3264", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["512", " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", FractionBox[SuperscriptBox["z", RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]]]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]]]]]]]], "]"]]]]]], "-", FractionBox[RowBox[List["69615", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["5", "/", "4"]]], " ", RowBox[List["(", RowBox[List["663", "+", RowBox[List["3536", " ", "z"]], "+", RowBox[List["3264", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["512", " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox[RowBox[List[SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]], "+", FractionBox[SqrtBox["z"], SqrtBox[RowBox[List["1", "-", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["37", "/", "4"]]]], "+", FractionBox[RowBox[List["69615", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["5", "/", "4"]]], " ", RowBox[List["(", RowBox[List["663", "+", RowBox[List["3536", " ", "z"]], "+", RowBox[List["3264", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["512", " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List[SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]], "+", FractionBox[SqrtBox["z"], SqrtBox[RowBox[List["1", "-", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["37", "/", "4"]]]]]], ")"]]]]]]]]

 MathML Form

 2 F 1 ( 4 , 5 ; - 1 4 ; z ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "2"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[RowBox[List[TagBox["4", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]], ",", TagBox["5", HypergeometricPFQ, Rule[Editable, True], Rule[Selectable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["-", FractionBox["1", "4"]]], 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 2097152 ( 139230 2 ( 512 z 3 + 3264 z 2 + 3536 z + 663 ) tan - 1 ( 1 - z 4 2 1 - z 4 , - z 4 2 1 - z 4 ) z 5 / 4 ( 1 - z ) 37 / 4 + 139230 2 ( 512 z 3 + 3264 z 2 + 3536 z + 663 ) tan - 1 ( z 4 2 1 - z 4 + 1 , - z 4 2 1 - z 4 ) z 5 / 4 ( 1 - z ) 37 / 4 - 69615 2 ( 512 z 3 + 3264 z 2 + 3536 z + 663 ) log ( z 1 - z - 2 z 4 1 - z 4 + 1 ) z 5 / 4 ( 1 - z ) 37 / 4 + 69615 2 ( 512 z 3 + 3264 z 2 + 3536 z + 663 ) log ( z 1 - z + 2 z 4 1 - z 4 + 1 ) z 5 / 4 ( 1 - z ) 37 / 4 + 8 ( 424320 z 5 + 59101856 z 4 + 266720156 z 3 + 205864621 z 2 + 23330816 z - 262144 ) ( z - 1 ) 9 ) HypergeometricPFQ 4 5 -1 1 4 z 1 2097152 139230 2 1 2 512 z 3 3264 z 2 3536 z 663 1 -1 z 1 4 2 1 2 1 -1 z 1 4 -1 -1 z 1 4 2 1 2 1 -1 z 1 4 -1 z 5 4 1 -1 z 37 4 -1 139230 2 1 2 512 z 3 3264 z 2 3536 z 663 z 1 4 2 1 2 1 -1 z 1 4 -1 1 -1 z 1 4 2 1 2 1 -1 z 1 4 -1 z 5 4 1 -1 z 37 4 -1 -1 69615 2 1 2 512 z 3 3264 z 2 3536 z 663 z 1 2 1 -1 z 1 2 -1 -1 2 1 2 z 1 4 1 -1 z 1 4 -1 1 z 5 4 1 -1 z 37 4 -1 69615 2 1 2 512 z 3 3264 z 2 3536 z 663 z 1 2 1 -1 z 1 2 -1 2 1 2 z 1 4 1 -1 z 1 4 -1 1 z 5 4 1 -1 z 37 4 -1 8 424320 z 5 59101856 z 4 266720156 z 3 205864621 z 2 23330816 z -262144 z -1 9 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Hypergeometric2F1", "[", RowBox[List["4", ",", "5", ",", RowBox[List["-", FractionBox["1", "4"]]], ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[FractionBox[RowBox[List["8", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "262144"]], "+", RowBox[List["23330816", " ", "z"]], "+", RowBox[List["205864621", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["266720156", " ", SuperscriptBox["z", "3"]]], "+", RowBox[List["59101856", " ", SuperscriptBox["z", "4"]]], "+", RowBox[List["424320", " ", SuperscriptBox["z", "5"]]]]], ")"]]]], SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]], "9"]], "+", FractionBox[RowBox[List["139230", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["5", "/", "4"]]], " ", RowBox[List["(", RowBox[List["663", "+", RowBox[List["3536", " ", "z"]], "+", RowBox[List["3264", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["512", " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "-", FractionBox[SuperscriptBox["z", RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]]]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["37", "/", "4"]]]], "+", FractionBox[RowBox[List["139230", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["5", "/", "4"]]], " ", RowBox[List["(", RowBox[List["663", "+", RowBox[List["3536", " ", "z"]], "+", RowBox[List["3264", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["512", " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", FractionBox[SuperscriptBox["z", RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]]]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox["z", RowBox[List["1", "/", "4"]]], RowBox[List[SqrtBox["2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]]]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["37", "/", "4"]]]], "-", FractionBox[RowBox[List["69615", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["5", "/", "4"]]], " ", RowBox[List["(", RowBox[List["663", "+", RowBox[List["3536", " ", "z"]], "+", RowBox[List["3264", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["512", " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "-", FractionBox[RowBox[List[SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]], "+", FractionBox[SqrtBox["z"], SqrtBox[RowBox[List["1", "-", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["37", "/", "4"]]]], "+", FractionBox[RowBox[List["69615", " ", SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["5", "/", "4"]]], " ", RowBox[List["(", RowBox[List["663", "+", RowBox[List["3536", " ", "z"]], "+", RowBox[List["3264", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["512", " ", SuperscriptBox["z", "3"]]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List["1", "+", FractionBox[RowBox[List[SqrtBox["2"], " ", SuperscriptBox["z", RowBox[List["1", "/", "4"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["1", "/", "4"]]]], "+", FractionBox[SqrtBox["z"], SqrtBox[RowBox[List["1", "-", "z"]]]]]], "]"]]]], SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["37", "/", "4"]]]]]], "2097152"]]]]]

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

 2007-05-02