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/01.30.16.0254.01

 Input Form

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

 Standard Form

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

 MathML Form

 a sech - 1 ( x ) + b csch - 1 ( y ) π ( 1 - ( - 1 ) - arg ( ( 1 + 1 y 2 + 1 y ) b ( 1 x - 1 1 + 1 x + 1 x ) a + 1 ) 2 π - - arg ( ( 1 x - 1 1 + 1 x + 1 x ) a ( 1 + 1 y 2 + 1 y ) b ) 2 π ) + ( - 1 ) arg ( ( 1 x - 1 1 + 1 x + 1 x ) a ( 1 + 1 y 2 + 1 y ) b ) π - 2 arg ( ( 1 x - 1 1 + 1 x + 1 x ) a ( 1 + 1 y 2 + 1 y ) b - 1 ) π + - arg ( ( 1 x - 1 1 + 1 x + 1 x ) a ( 1 + 1 y 2 + 1 y ) b ) π - - arg ( ( 1 x - 1 1 + 1 x + 1 x ) a ( 1 + 1 y 2 + 1 y ) b - 1 ) 2 π + - arg ( ( 1 + 1 y 2 + 1 y ) b ( 1 x - 1 1 + 1 x + 1 x ) a + 1 ) 2 π sech - 1 ( 2 ( 1 x - 1 1 + 1 x + 1 x ) a ( 1 + 1 y 2 + 1 y ) b ( 1 + 1 y 2 + 1 y ) 2 b ( 1 x - 1 1 + 1 x + 1 x ) 2 a + 1 ) - 2 π ( - arg ( ( 1 x - 1 1 + 1 x + 1 x ) a ) - arg ( ( 1 + 1 y 2 + 1 y ) b ) + π 2 π + π - Im ( a log ( 1 x - 1 1 + 1 x + 1 x ) ) 2 π + π - Im ( b log ( 1 + 1 y 2 + 1 y ) ) 2 π ) a x b y 1 -1 -1 -1 1 1 y 2 -1 1 2 1 y -1 b 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 a 1 2 -1 -1 -1 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 a 1 1 y 2 -1 1 2 1 y -1 b 2 -1 -1 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 a 1 1 y 2 -1 1 2 1 y -1 b -1 -1 2 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 a 1 1 y 2 -1 1 2 1 y -1 b -1 -1 -1 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 a 1 1 y 2 -1 1 2 1 y -1 b -1 -1 -1 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 a 1 1 y 2 -1 1 2 1 y -1 b -1 2 -1 -1 1 1 y 2 -1 1 2 1 y -1 b 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 a 1 2 -1 2 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 a 1 1 y 2 -1 1 2 1 y -1 b 1 1 y 2 -1 1 2 1 y -1 2 b 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 2 a 1 -1 -1 2 -1 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 a -1 1 1 y 2 -1 1 2 1 y -1 b 2 -1 -1 a 1 x -1 -1 1 2 1 1 x -1 1 2 1 x -1 2 -1 -1 b 1 1 y 2 -1 1 2 1 y -1 2 -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["a_", " ", RowBox[List["ArcSech", "[", "x_", "]"]]]], "+", RowBox[List["b_", " ", RowBox[List["ArcCsch", "[", "y_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[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[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], "+", FractionBox["1", "y"]]], ")"]], "b"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[FractionBox["1", "x"], "-", "1"]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "x"]]]]]], "+", FractionBox["1", "x"]]], ")"]], "a"]]], "+", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]], "-", RowBox[List["Floor", "[", 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02