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ArcCoth






Mathematica Notation

Traditional Notation









Elementary Functions > ArcCoth[z] > Series representations > Generalized power series > Expansions at z==-1 > For small integer powers of the function > For the second power





http://functions.wolfram.com/01.28.06.0055.01









  


  










Input Form





ArcCoth[z]^2 \[Proportional] (1/4) Log[-((z + 1)/2)]^2 + (1/4) Log[-((z + 1)/2)] (z + 1) (1 + (z + 1)/4 + (z + 1)^2/12 + \[Ellipsis]) + (1/16) (z + 1)^2 (1 + (z + 1)/2 + (11/48) (z + 1)^2 + \[Ellipsis]) /; (z -> -1)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["ArcCoth", "[", "z", "]"]], "2"], "\[Proportional]", RowBox[List[RowBox[List[FractionBox["1", "4"], SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", FractionBox[RowBox[List["z", "+", "1"]], "2"]]], "]"]], "2"]]], "+", RowBox[List[FractionBox["1", "4"], RowBox[List["Log", "[", RowBox[List["-", FractionBox[RowBox[List["z", "+", "1"]], "2"]]], "]"]], RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["z", "+", "1"]], "4"], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"], "12"], " ", "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[FractionBox["1", "16"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["z", "+", "1"]], "2"], "+", RowBox[List[FractionBox["11", "48"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["-", "1"]]]], ")"]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mrow> <msup> <mi> coth </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> &#8733; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 4 </mn> </mfrac> <mo> + </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mn> 12 </mn> </mfrac> <mtext> </mtext> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mtext> </mtext> <mo> + </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mn> 16 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mrow> <mfrac> <mn> 11 </mn> <mn> 48 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mtext> </mtext> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <power /> <apply> <arccoth /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <power /> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ln /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 12 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 16 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 11 <sep /> 48 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["ArcCoth", "[", "z_", "]"]], "2"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]], "]"]], "2"]]], "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]]]], "]"]], " ", RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["z", "+", "1"]], "4"], "+", RowBox[List[FractionBox["1", "12"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]], "+", RowBox[List[FractionBox["1", "16"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["z", "+", "1"]], "2"], "+", RowBox[List[FractionBox["11", "48"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", "1"]], ")"]], "2"]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["-", "1"]]]], ")"]]]]]]]]










Date Added to functions.wolfram.com (modification date)





2007-05-02





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