html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 ArcSech

 http://functions.wolfram.com/01.30.16.0189.01

 Input Form

 ArcSech[x] + ArcCos[y] == -2 I Pi (Floor[(-Arg[Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x] - Arg[(I y + Sqrt[1 - y^2])^I] + Pi)/(2 Pi)] + Floor[(Pi - Im[Log[Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x]])/(2 Pi)] + Floor[(Pi - Re[Log[I y + Sqrt[1 - y^2]]])/(2 Pi)]) + I Pi (1 - (-1)^(Floor[-(Arg[(I y + Sqrt[1 - y^2])^I (Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x) + 1]/(2 Pi))] - Floor[-(Arg[(Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x) (I y + Sqrt[1 - y^2])^ I]/(2 Pi))])) + (-1)^(Floor[-(Arg[(Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x) (I y + Sqrt[1 - y^2])^ I]/Pi)] - Floor[-(Arg[(Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x) (I y + Sqrt[1 - y^2])^ I - 1]/(2 Pi))] + Floor[Arg[(Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x) (I y + Sqrt[1 - y^2])^I]/ Pi - (2 Arg[(Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x) (I y + Sqrt[1 - y^2])^ I - 1])/Pi] + Floor[-(Arg[(I y + Sqrt[1 - y^2])^I (Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x) + 1]/(2 Pi))]) ArcSech[(2 (Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x) (I y + Sqrt[1 - y^2])^I)/ ((I y + Sqrt[1 - y^2])^(2 I) (Sqrt[1/x - 1] Sqrt[1 + 1/x] + 1/x)^2 + 1)] + Pi/2

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSech", "[", "x", "]"]], "+", RowBox[List["ArcCos", "[", "y", "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", RowBox[List[RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "x"], "-", "1"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "x"]]]]]], "+", FractionBox["1", "x"]]], "]"]]]], "-", RowBox[List["Arg", "[", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], ")"]], "\[ImaginaryI]"], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Im", "[", RowBox[List["Log", "[", RowBox[List[RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "x"], "-", "1"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "x"]]]]]], "+", FractionBox["1", "x"]]], "]"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Re", "[", RowBox[List["Log", "[", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], "]"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List["Arg", "[", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], ")"]], "\[ImaginaryI]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "x"], "-", "1"]]], " ", 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RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], ")"]], "\[ImaginaryI]"], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "x"], "-", "1"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "x"]]]]]], "+", FractionBox["1", "x"]]], ")"]]]], "+", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]], " ", RowBox[List["ArcSech", "[", FractionBox[RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "x"], "-", "1"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "x"]]]]]], "+", FractionBox["1", "x"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], ")"]], "\[ImaginaryI]"]]], RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], ")"]], RowBox[List["2", " ", "\[ImaginaryI]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "x"], "-", "1"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "x"]]]]]], "+", FractionBox["1", "x"]]], ")"]], "2"]]], "+", "1"]]], "]"]]]], "+", FractionBox["\[Pi]", "2"]]]]]]]

 MathML Form

 sech - 1 ( x ) + cos - 1 ( y ) - 2 π ( - arg ( 1 x - 1 1 + 1 x + 1 x ) - arg ( ( y + 1 - y 2 ) ) + π 2 π + π - Im ( log ( 1 x - 1 1 + 1 x + 1 x ) ) 2 π + π - Re ( log ( y + 1 - y 2 ) ) 2 π ) + π ( 1 - ( - 1 ) - arg ( ( 1 x - 1 1 + 1 x + 1 x ) ( y + 1 - y 2 ) + 1 ) 2 π - - arg ( ( 1 x - 1 1 + 1 x + 1 x ) ( y + 1 - y 2 ) ) 2 π ) + ( - 1 ) arg ( ( 1 x - 1 1 + 1 x + 1 x ) ( y + 1 - y 2 ) ) π - 2 arg ( ( 1 x - 1 1 + 1 x + 1 x ) ( y + 1 - y 2 ) - 1 ) π + - arg ( ( 1 x - 1 1 + 1 x + 1 x ) ( y + 1 - y 2 ) ) π - - arg ( ( 1 x - 1 1 + 1 x + 1 x ) ( y + 1 - y 2 ) - 1 ) 2 π + - arg ( ( 1 x - 1 1 + 1 x + 1 x ) ( y + 1 - y 2 ) + 1 ) 2 π sech - 1 ( 2 ( 1 x - 1 1 + 1 x + 1 x ) ( y + 1 - y 2 ) ( 1 x - 1 1 + 1 x + 1 x ) 2 ( y + 1 - y 2 ) 2 + 1 ) + π 2 x y -2 -1 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 -1 y 1 -1 y 2 1 2 2 -1 -1 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 2 -1 -1 y 1 -1 y 2 1 2 2 -1 1 -1 -1 -1 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 y 1 -1 y 2 1 2 1 2 -1 -1 -1 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 y 1 -1 y 2 1 2 2 -1 -1 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 y 1 -1 y 2 1 2 -1 -1 2 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 y 1 -1 y 2 1 2 -1 -1 -1 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 y 1 -1 y 2 1 2 -1 -1 -1 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 y 1 -1 y 2 1 2 -1 2 -1 -1 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 y 1 -1 y 2 1 2 1 2 -1 2 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 y 1 -1 y 2 1 2 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 2 y 1 -1 y 2 1 2 2 1 -1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["ArcSech", "[", "x_", "]"]], "+", RowBox[List["ArcCos", "[", "y_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", RowBox[List[RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "x"], "-", "1"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "x"]]]]]], "+", FractionBox["1", "x"]]], "]"]]]], "-", RowBox[List["Arg", "[", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], ")"]], "\[ImaginaryI]"], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Im", "[", RowBox[List["Log", "[", 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RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "y"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["y", "2"]]]]]], ")"]], RowBox[List["2", " ", "\[ImaginaryI]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "x"], "-", "1"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "x"]]]]]], "+", FractionBox["1", "x"]]], ")"]], "2"]]], "+", "1"]]], "]"]]]], "+", FractionBox["\[Pi]", "2"]]]]]]]

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

 2007-05-02