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http://functions.wolfram.com/01.29.19.0004.01
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Arg[ArcCsch[x + I y]] ==
ArcTan[
Log[Sqrt[(x/(x^2 + y^2) + ((x^4 + (-1 + y^2)^2 + 2 x^2 (1 + y^2))/
(x^2 + y^2)^2)^(1/4)
Cos[(1/2) ArcTan[1 + (x^2 - y^2)/(x^2 + y^2)^2,
-((2 x y)/(x^2 + y^2)^2)]])^2 +
(y/(x^2 + y^2) - ((x^4 + (-1 + y^2)^2 + 2 x^2 (1 + y^2))/(x^2 + y^2)^2)^
(1/4) Sin[(1/2) ArcTan[1 + (x^2 - y^2)/(x^2 + y^2)^2,
-((2 x y)/(x^2 + y^2)^2)]])^2]],
ArcTan[(1/(x^2 + y^2)) (x + (x^2 + y^2)
((x^4 + (-1 + y^2)^2 + 2 x^2 (1 + y^2))/(x^2 + y^2)^2)^(1/4)
Cos[(1/2) ArcTan[1 + (x^2 - y^2)/(x^2 + y^2)^2,
-((2 x y)/(x^2 + y^2)^2)]]), (1/(x^2 + y^2))
(-y + (x^2 + y^2) ((x^4 + (-1 + y^2)^2 + 2 x^2 (1 + y^2))/(x^2 + y^2)^2)^
(1/4) Sin[(1/2) ArcTan[1 + (x^2 - y^2)/(x^2 + y^2)^2,
-((2 x y)/(x^2 + y^2)^2)]])]]
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Cell[BoxData[RowBox[List[RowBox[List["Arg", "[", RowBox[List["ArcCsch", "[", RowBox[List["x", "+", RowBox[List["\[ImaginaryI]", " ", "y"]]]], "]"]], "]"]], "\[Equal]", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["Log", "[", RowBox[List["\[Sqrt]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["x", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[SuperscriptBox["x", "4"], "+", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["y", "2"]]], ")"]], "2"], "+", RowBox[List["2", " ", SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox["y", "2"]]], ")"]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], ")"]], "2"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox["1", "2"], " ", 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SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], ")"]], "2"]], ")"]], RowBox[List["1", "/", "4"]]], " ", RowBox[List["Sin", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List[SuperscriptBox["x", "2"], "-", SuperscriptBox["y", "2"]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], ")"]], "2"]]]], ",", RowBox[List["-", FractionBox[RowBox[List["2", " ", "x", " ", "y"]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], ")"]], "2"]]]]]], "]"]]]], "]"]]]]]], ")"]], "^", "2"]]]], ")"]]]], "]"]], ",", RowBox[List["ArcTan", "[", RowBox[List[RowBox[List[FractionBox["1", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]]], RowBox[List["(", RowBox[List["x", "+", RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["x", "2"], "+", 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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> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <msup> <mi> csch </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> √ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mroot> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mn> 4 </mn> </mroot> </mrow> <mo> + </mo> <mfrac> <mi> x </mi> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ^ </mo> <mn> 2 </mn> </mrow> <mo> + </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mi> y </mi> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> - </mo> <mrow> <mroot> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ^ </mo> <mn> 2 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> , </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> x </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mroot> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mn> 4 </mn> </mroot> </mrow> <mo> + </mo> <mi> x </mi> </mrow> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> , </mo> <mfrac> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mroot> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mn> 4 </mn> </mroot> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mrow> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <msup> <mrow> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mfrac> <mo> + </mo> <mn> 1 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Date Added to functions.wolfram.com (modification date)
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