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http://functions.wolfram.com/01.28.27.3550.01
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ArcCoth[Tanh[z]] == z - (I Pi)/2 - I Pi Floor[Im[z]/Pi] +
((Pi I)/2) (1 + (-1)^(-Ceiling[Im[z]/Pi] + Floor[Im[z]/Pi]))
UnitStep[-Re[z]] /; !Element[(I z)/Pi, Integers]
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCoth", "[", RowBox[List["Tanh", "[", "z", "]"]], "]"]], "\[Equal]", RowBox[List["z", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"], RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", RowBox[List["Ceiling", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]]]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", "z", "]"]]]], "]"]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List["Element", "[", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Pi]"], ",", "Integers"]], "]"]], "]"]]]]]]
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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> coth </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> tanh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mi> z </mi> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> π </mi> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mn> 2 </mn> </mfrac> <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> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> π </mi> </mfrac> <mo> ⌋ </mo> </mrow> <mo> - </mo> <mrow> <mo> ⌈ </mo> <mfrac> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> π </mi> </mfrac> <mo> ⌉ </mo> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <semantics> <mi> θ </mi> <annotation-xml encoding='MathML-Content'> <ci> UnitStep </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mi> π </mi> </mfrac> <mo> ∉ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arccoth /> <apply> <tanh /> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> <apply> <floor /> <apply> <times /> <apply> <imaginary /> <ci> z </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <floor /> <apply> <times /> <apply> <imaginary /> <ci> z </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ceiling /> <apply> <times /> <apply> <imaginary /> <ci> z </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> UnitStep </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <notin /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCoth", "[", RowBox[List["Tanh", "[", "z_", "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["z", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", RowBox[List["Ceiling", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]]]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List["-", RowBox[List["Re", "[", "z", "]"]]]], "]"]]]]]], "/;", RowBox[List["!", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Pi]"], "\[Element]", "Integers"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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