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http://functions.wolfram.com/01.29.16.0189.01
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ArcCsch[x] + ArcCoth[y] ==
(-((I x Sqrt[1 - 1/y^2] y Sqrt[-((1 + x^2 + 2 Sqrt[1 + 1/x^2] x y + y^2)/
(x^2 (-1 + y^2)))])/(Sqrt[1 + 1/x^2] x + y)))
ArcSinh[(I Sqrt[1 + 1/x^2] x + I/y)/(x Sqrt[1 - 1/y^2])] +
(Pi (Sqrt[1 + 1/x^2] x + y))/(2 x Sqrt[1 - 1/y^2] y
Sqrt[-((1 + x^2 + 2 Sqrt[1 + 1/x^2] x y + y^2)/(x^2 (-1 + y^2)))]) +
Pi I (1 - (I (Sqrt[1 + 1/x^2] x + y))/(x Sqrt[1 - 1/y^2] y
Sqrt[-((1 + x^2 + 2 Sqrt[1 + 1/x^2] x y + y^2)/(x^2 (-1 + y^2)))]))
Floor[(Arg[Sqrt[1 + 1/x^2] + 1/x] + Arg[(I + I/y)/Sqrt[1 - 1/y^2]])/
(2 Pi)] -
Pi I (1 + (x Sqrt[1 - 1/y^2] y
Sqrt[-((-1 - x^2 - 2 Sqrt[1 + 1/x^2] x y - y^2)/(x^2 (1 - y^2)))])/
(I Sqrt[1 + 1/x^2] x + I y))
Floor[-((-Pi + Arg[Sqrt[1 + 1/x^2] + 1/x] +
Arg[(I + I/y)/Sqrt[1 - 1/y^2]])/(2 Pi))]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCsch", "[", "x", "]"]], "+", RowBox[List["ArcCoth", "[", "y", "]"]]]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "x", " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], " ", "y", " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["1", "+", SuperscriptBox["x", "2"], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", "y"]], "+", SuperscriptBox["y", "2"]]], RowBox[List[SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["y", "2"]]], ")"]]]]]]]]]], RowBox[List[RowBox[List[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"]], "+", FractionBox["\[ImaginaryI]", "y"]]], RowBox[List["x", " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]]]]], "]"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x"]], "+", "y"]], ")"]]]], RowBox[List["2", " ", "x", " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], " ", "y", " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["1", "+", SuperscriptBox["x", "2"], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", "y"]], "+", SuperscriptBox["y", "2"]]], RowBox[List[SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["y", "2"]]], ")"]]]]]]]]]]], "+", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x"]], "+", "y"]], ")"]]]], RowBox[List["x", " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], " ", "y", " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["1", "+", SuperscriptBox["x", "2"], "+", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", "y"]], "+", SuperscriptBox["y", "2"]]], RowBox[List[SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List[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]", "+", FractionBox["\[ImaginaryI]", "y"]]], SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["x", " ", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["y", "2"]]]]], " ", "y", " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["x", "2"], "-", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x", " ", "y"]], "-", SuperscriptBox["y", "2"]]], RowBox[List[SuperscriptBox["x", "2"], " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["y", "2"]]], ")"]]]]]]]]]], RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["x", "2"]]]]], " ", "x"]], "+", RowBox[List["\[ImaginaryI]", " ", "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]", "+", FractionBox["\[ImaginaryI]", "y"]]], SqrtBox[RowBox[List["1", "-", FractionBox["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> <msup> <mi> coth </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo>  </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <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> <mo> ⁢ </mo> <mi> y </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <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> y </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <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> </mrow> <mrow> <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> <mi> x </mi> </mrow> <mo> + </mo> <mi> y </mi> </mrow> </mfrac> </mrow> <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> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </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> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <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> <mi> x </mi> </mrow> <mo> + </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <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> <mo> ⁢ </mo> <mi> y </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <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> y </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <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> </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> <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> <mi> x </mi> </mrow> <mo> + </mo> <mi> y </mi> </mrow> <mo> ) </mo> </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> <mo> ⁢ </mo> <mi> y </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <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> y </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <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> </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> <mfrac> <mi> ⅈ </mi> <mi> y </mi> </mfrac> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </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> x </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mrow> <mrow> <mo> - </mo> <msup> <mi> x </mi> <mn> 2 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <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> y </mi> <mo> ⁢ </mo> <mi> x </mi> </mrow> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> y </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> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> y </mi> </mrow> </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> 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Date Added to functions.wolfram.com (modification date)
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