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http://functions.wolfram.com/01.14.16.0221.01
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ArcTan[x] + I ArcCsch[y] ==
(-((Sqrt[1 + x^2] Sqrt[(x Sqrt[1 + 1/y^2] + I/y)^2/(1 + x^2)])/
(x Sqrt[1 + 1/y^2] + I/y))) ArcSin[(Sqrt[1 + 1/y^2] - (I x)/y)/
Sqrt[1 + x^2]] +
(Pi Sqrt[1 + x^2] Sqrt[(x Sqrt[1 + 1/y^2] + I/y)^2/(1 + x^2)])/
(2 (x Sqrt[1 + 1/y^2] + I/y)) +
Pi (1 + (Sqrt[1 + x^2] Sqrt[(x Sqrt[1 + 1/y^2] + I/y)^2/(1 + x^2)])/
(x Sqrt[1 + 1/y^2] + I/y))
Floor[(Arg[(I - x)/Sqrt[1 + x^2]] + Arg[Sqrt[1 + 1/y^2] - 1/y])/(2 Pi)] -
Pi (-1 + (Sqrt[1 + x^2] Sqrt[(x Sqrt[1 + 1/y^2] + I/y)^2/(1 + x^2)])/
(x Sqrt[1 + 1/y^2] + I/y))
Floor[-((-Pi + Arg[(I - x)/Sqrt[1 + x^2]] + Arg[Sqrt[1 + 1/y^2] - 1/y])/
(2 Pi))]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcTan", "[", "x", "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcCsch", "[", "y", "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]], ")"]], "2"], RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]], " "]], RowBox[List[RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]]]]], RowBox[List["ArcSin", "[", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "x"]], "y"]]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]], "]"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]], ")"]], "2"], RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]]]], RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]], ")"]]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]], ")"]], "2"], RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]]]], RowBox[List[RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "-", "x"]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], "-", FractionBox["1", "y"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", FractionBox[RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]], " ", SqrtBox[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]], ")"]], "2"], RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]]]], RowBox[List[RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]]]], "+", FractionBox["\[ImaginaryI]", "y"]]]]]], ")"]], " ", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", "\[Pi]"]], "+", RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "-", "x"]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["x", "2"]]]]], "]"]], "+", RowBox[List["Arg", "[", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], "-", FractionBox["1", "y"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]]]]]]]
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<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> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo>  </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <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> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mtext> </mtext> </mrow> <mrow> <mrow> <mi> x </mi> <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> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> sin </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <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> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mi> y </mi> </mfrac> </mrow> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msqrt> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <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> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <mrow> <mi> x </mi> <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> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> - </mo> <mi> x </mi> </mrow> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> arg </mi> <mo> ⁡ </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> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <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> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> </mrow> <mrow> <mrow> <mi> x </mi> <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> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> - </mo> <mi> x </mi> </mrow> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> arg </mi> <mo> ⁡ </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> <mn> 1 </mn> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> ⌋ </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msqrt> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <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> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> x </mi> <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> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <arctan /> <ci> x </ci> </apply> <apply> <times /> <imaginaryi /> <apply> <arccsch /> <ci> y </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <plus /> <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> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <plus /> <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> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <arcsin /> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <plus /> <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> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> x </ci> <apply> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <pi /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> x </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn 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Date Added to functions.wolfram.com (modification date)
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