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http://functions.wolfram.com/01.27.27.3489.01
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ArcTanh[Coth[z]] == Piecewise[{{Indeterminate, Element[(I z)/Pi, Integers]},
{z - Pi I Floor[Im[z]/Pi] + (I Pi)/2, Element[Im[z]/Pi, Integers] &&
Re[z] < 0}, {z - Pi I Floor[Im[z]/Pi] - (I Pi)/2, True}}]
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Cell[BoxData[RowBox[List[RowBox[List["ArcTanh", "[", RowBox[List["Coth", "[", "z", "]"]], "]"]], "\[Equal]", RowBox[List["Piecewise", "[", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["Indeterminate", ",", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Pi]"], "\[Element]", "Integers"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List["z", "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"]]], ",", RowBox[List[RowBox[List[FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "<", "0"]]]]]], "}"]], ",", "\[IndentingNewLine]", RowBox[List["{", RowBox[List[RowBox[List["z", "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"]]], ",", "True"]], "}"]]]], "}"]], "]"]]]]]]
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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> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> coth </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mo>  </mo> <mtable> <mtr> <mtd> <mi> ¿ </mi> </mtd> <mtd> <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> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </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> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mtd> <mtd> <mrow> <mrow> <mfrac> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> π </mi> </mfrac> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </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> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> </mtd> <mtd> <mi> True </mi> </mtd> </mtr> </mtable> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arctanh /> <apply> <coth /> <ci> z </ci> </apply> </apply> <apply> <ci> Piecewise </ci> <list> <list> <ci> ¿ </ci> <apply> <in /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> </list> <list> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> <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 /> <imaginaryi /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <times /> <apply> <imaginary /> <ci> z </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> <apply> <lt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </list> <list> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> <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 /> <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> <true /> </list> </list> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTanh", "[", RowBox[List["Coth", "[", "z_", "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["\[Piecewise]", GridBox[List[List["Indeterminate", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Pi]"], "\[Element]", "Integers"]]], List[RowBox[List["z", "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"]]], RowBox[List[RowBox[List[FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "<", "0"]]]]], List[RowBox[List["z", "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"], "]"]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"]]], "True"]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
