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http://functions.wolfram.com/01.16.16.0257.01
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ArcCot[x] - ArcCsch[y] ==
Pi Floor[(-Arg[1 - I/x] + Arg[1 + I/x] + Pi)/(2 Pi)] -
2 I Pi (Floor[(-Arg[((x - I)/(x + I))^(I/2)] -
Arg[1/(Sqrt[1 + 1/y^2] + 1/y)] + Pi)/(2 Pi)] +
Floor[(Im[Log[Sqrt[1 + 1/y^2] + 1/y]] + Pi)/(2 Pi)] +
Floor[(Pi - (1/2) Re[Log[(x - I)/(x + I)]])/(2 Pi)]) +
I Pi (1 - (-1)^Floor[Arg[((x - I)/(x + I))^(I/2)/(Sqrt[1 + 1/y^2] + 1/y) +
1]/(2 Pi) + 1/2]) +
2 I ArcCot[(I (((x - I)/(x + I))^(I/2)/(Sqrt[1 + 1/y^2] + 1/y) + 1))/
(((x - I)/(x + I))^(I/2)/(Sqrt[1 + 1/y^2] + 1/y) - 1)]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCot", "[", "x", "]"]], "-", RowBox[List["ArcCsch", "[", "y", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", RowBox[List["1", "-", FractionBox["\[ImaginaryI]", "x"]]], "]"]]]], "+", RowBox[List["Arg", "[", RowBox[List["1", "+", FractionBox["\[ImaginaryI]", "x"]]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["-", RowBox[List["Arg", "[", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["x", "-", "\[ImaginaryI]"]], RowBox[List["x", "+", "\[ImaginaryI]"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], "]"]]]], "-", RowBox[List["Arg", "[", FractionBox["1", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], "+", FractionBox["1", "y"]]]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Im", "[", RowBox[List["Log", "[", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], "+", FractionBox["1", "y"]]], "]"]], "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Re", "[", RowBox[List["Log", "[", FractionBox[RowBox[List["x", "-", "\[ImaginaryI]"]], RowBox[List["x", "+", "\[ImaginaryI]"]]], "]"]], "]"]]]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox[RowBox[List["Arg", "[", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["x", "-", "\[ImaginaryI]"]], RowBox[List["x", "+", "\[ImaginaryI]"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], "+", FractionBox["1", "y"]]]], "+", "1"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "+", FractionBox["1", "2"]]], "]"]]]]], ")"]]]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcCot", "[", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["x", "-", "\[ImaginaryI]"]], RowBox[List["x", "+", "\[ImaginaryI]"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], "+", FractionBox["1", "y"]]]], "+", "1"]], ")"]]]], RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List["x", "-", "\[ImaginaryI]"]], RowBox[List["x", "+", "\[ImaginaryI]"]]], ")"]], RowBox[List["\[ImaginaryI]", "/", "2"]]], RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", SuperscriptBox["y", "2"]]]]], "+", FractionBox["1", "y"]]]], "-", "1"]]], "]"]]]]]]]]]]
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<mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mrow> <mo> - </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> x </mi> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mrow> <mi> x </mi> <mo> + </mo> <mi> ⅈ </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> ⅈ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <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> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> log </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> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> π </mi> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> x </mi> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mrow> <mi> x </mi> <mo> + </mo> <mi> ⅈ </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </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> ⌊ </mo> <mrow> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> x </mi> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mrow> <mi> x </mi> <mo> + </mo> <mi> ⅈ </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> ⅈ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <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> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⌋ </mo> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> x </mi> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mrow> <mi> x </mi> <mo> + </mo> <mi> ⅈ </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> ⅈ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <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> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mfrac> <mrow> <mi> x </mi> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mrow> <mi> x </mi> <mo> + </mo> <mi> ⅈ </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mrow> <mi> ⅈ </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <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> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <arccot /> <ci> x </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arccsch /> <ci> y </ci> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <arg /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> x </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <pi /> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <pi /> <apply> <plus /> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> x </ci> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arg /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <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> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <pi /> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <imaginary /> <apply> <ln /> <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> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <pi /> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <real /> <apply> <ln /> <apply> <times /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> x </ci> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </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> <times /> <imaginaryi /> <pi /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <plus /> <apply> <times /> <apply> <arg /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> x </ci> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <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> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <arccot /> <apply> <times /> <imaginaryi /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> x </ci> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <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> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> x </ci> <imaginaryi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <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> <power /> <ci> y </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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Date Added to functions.wolfram.com (modification date)
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