ArcTanh[x] - I ArcCsc[y] == -ArcTanh[(Sqrt[1 - x^2] Sqrt[1 - ((I x)/y - Sqrt[1 - 1/y^2])^2/(1 - x^2)])/ ((-1)^Floor[1/2 - Arg[((-I) Sqrt[1 - 1/y^2] x - 1/y)/Sqrt[1 - x^2]]/Pi] (x/y + I Sqrt[1 - 1/y^2]))] + (1/2) I Pi (2 (1 + (-1)^Floor[1/2 - Arg[((-I) Sqrt[1 - 1/y^2] x - 1/y)/Sqrt[1 - x^2]]/ Pi]) Floor[(Arg[(I x + I)/Sqrt[1 - x^2]] + Arg[Sqrt[1 - 1/y^2] - I/y])/(2 Pi)] + (-1)^Floor[1/2 - Arg[((-I) Sqrt[1 - 1/y^2] x - 1/y)/Sqrt[1 - x^2]]/Pi] + (-1)^(Floor[1/2 - Arg[((-I) Sqrt[1 - 1/y^2] x - 1/y)/Sqrt[1 - x^2]]/Pi] + Floor[-(Arg[1 - x^2]/(2 Pi)) + Arg[(I x)/y - Sqrt[1 - 1/y^2]]/Pi + 1/2]) - 2 (-1 + (-1)^Floor[1/2 - Arg[((-I) Sqrt[1 - 1/y^2] x - 1/y)/ Sqrt[1 - x^2]]/Pi]) Floor[1/2 - (Arg[(I x + I)/Sqrt[1 - x^2]] + Arg[Sqrt[1 - 1/y^2] - I/y])/ (2 Pi)])

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTanh", "[", "x", "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcCsc", "[", "y", "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["-", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], " ", "x"]], "-", FractionBox["1", "y"]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]], "\[Pi]"]]], "]"]]]]], " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]], " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "x"]], "y"], "-", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], ")"]], "2"], RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]]]]]], RowBox[List[FractionBox["x", "y"], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]]]]]]], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], " ", "x"]], "-", FractionBox["1", "y"]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]], "\[Pi]"]]], "]"]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "\[ImaginaryI]"]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], "-", FractionBox["\[ImaginaryI]", "y"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], " ", "x"]], "-", FractionBox["1", "y"]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]], "\[Pi]"]]], "]"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], " ", "x"]], "-", FractionBox["1", "y"]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]], "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["Arg", "[", RowBox[List["1", "-", SuperscriptBox["x", "2"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "+", FractionBox[RowBox[List["Arg", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "x"]], "y"], "-", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "]"]], "\[Pi]"], "+", FractionBox["1", "2"]]], "]"]]]]], "-", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], " ", "x"]], "-", FractionBox["1", "y"]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]], "\[Pi]"]]], "]"]]]]], ")"]], " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List[RowBox[List["Arg", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", "x"]], "+", "\[ImaginaryI]"]], SqrtBox[RowBox[List["1", "-", SuperscriptBox["x", "2"]]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], "-", FractionBox["\[ImaginaryI]", "y"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]], ")"]]]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mfrac> <mrow> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <mi> &#8520; </mi> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> - </mo> <mfrac> <mi> &#8520; </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> &#8970; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mi> y </mi> </mfrac> <mo> - </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> <mo> + </mo> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> </mrow> </msup> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> </msup> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> + </mo> <mi> &#8520; </mi> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> - </mo> <mfrac> <mi> &#8520; </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mrow> <mo> &#8970; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mrow> <mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mi> &#960; </mi> </mfrac> </mrow> <mo> &#8971; </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> x </mi> </mrow> <mi> y </mi> </mfrac> <mo> - </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> </msqrt> </mrow> <mrow> <mfrac> <mi> x </mi> <mi> y </mi> </mfrac> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> </mrow> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <arctanh /> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <arccsc /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <imaginaryi /> <pi /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <arg /> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> <imaginaryi /> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <arg /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <floor /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <arg /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <floor /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <floor /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <apply> <arg /> <apply> <times /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> </apply> <imaginaryi /> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <arg /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arctanh /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <apply> <times /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> x </ci> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> x </ci> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

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

2007-05-02