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http://functions.wolfram.com/01.29.16.0204.01
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ArcCsch[x] - I ArcTan[y] ==
((I x Sqrt[-((-1 + I Sqrt[1 + 1/x^2] x y)^2/(x^2 (1 + y^2)))] Sqrt[1 + y^2])/
(-1 + I Sqrt[1 + 1/x^2] x y)) ArcSinh[(I Sqrt[1 + 1/x^2] x + y)/
(x Sqrt[1 + y^2])] + (Pi (1 - I Sqrt[1 + 1/x^2] x y))/
(2 x Sqrt[-((-1 + I Sqrt[1 + 1/x^2] x y)^2/(x^2 (1 + y^2)))]
Sqrt[1 + y^2]) + Pi I (1 - (I (1 - I Sqrt[1 + 1/x^2] x y))/
(x Sqrt[-((-1 + I Sqrt[1 + 1/x^2] x y)^2/(x^2 (1 + y^2)))]
Sqrt[1 + y^2])) Floor[(Arg[Sqrt[1 + 1/x^2] + 1/x] +
Arg[(I + y)/Sqrt[1 + y^2]])/(2 Pi)] -
Pi I (1 + (I x Sqrt[-((-1 + I Sqrt[1 + 1/x^2] x y)^2/(x^2 (1 + y^2)))]
Sqrt[1 + y^2])/(-1 + I Sqrt[1 + 1/x^2] x y))
Floor[-((-Pi + Arg[Sqrt[1 + 1/x^2] + 1/x] + Arg[(I + y)/Sqrt[1 + y^2]])/
(2 Pi))]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCsch", "[", "x", "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", "y", "]"]]]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", "y"]]]], ")"]], "2"], RowBox[List[SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["y", "2"]]], ")"]]]]]]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", "y"]]]]], RowBox[List["ArcSinh", "[", FractionBox[RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x"]], "+", "y"]], RowBox[List["x", " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]], "]"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", "y"]]]], ")"]]]], RowBox[List["2", " ", "x", " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", "y"]]]], ")"]], "2"], RowBox[List[SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["y", "2"]]], ")"]]]]]]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", "y"]]]], ")"]]]], RowBox[List["x", " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", "y"]]]], ")"]], "2"], RowBox[List[SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["y", "2"]]], ")"]]]]]]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]]]]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Arg", "[", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], "+", FractionBox["1", "x"]]], "]"]], "+", RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "+", "y"]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", "y"]]]], ")"]], "2"], RowBox[List[SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["y", "2"]]], ")"]]]]]]]], " ", SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]]], RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", "y"]]]]]]], ")"]], " ", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", "\[Pi]"]], "+", RowBox[List["Arg", "[", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], "+", FractionBox["1", "x"]]], "]"]], "+", RowBox[List["Arg", "[", FractionBox[RowBox[List["\[ImaginaryI]", "+", "y"]], SqrtBox[RowBox[List["1", "+", SuperscriptBox["y", "2"]]]]], "]"]]]], 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> csch </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> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo>  </mo> <mrow> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <mi> y </mi> </mrow> <mrow> <mi> x </mi> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> x </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> + </mo> <mi> y </mi> </mrow> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <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> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </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> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> x </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> + </mo> <mi> y </mi> </mrow> <msqrt> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <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> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <arccsch /> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <arctan /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <imaginaryi /> <ci> x </ci> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> 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Date Added to functions.wolfram.com (modification date)
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