Inverse hyperbolic cotangent: Series representations (formula 01.28.06.0064)

ArcCoth

$Mathematica Notation$

Elementary Functions

ArcCoth[z]

Series representations

Generalized power series

Expansions at z==infinity

For small integer powers of the function

For the second power

http://functions.wolfram.com/01.28.06.0064.01

Input Form

ArcCoth[z]^2 \[Proportional] (1/z^2) (1 + 2/(3 z^2) + 23/(45 z^4) + O[1/z^6])

Standard Form

Cell[BoxData[RowBox[List[SuperscriptBox[RowBox[List["ArcCoth", "[", "z", "]"]], "2"], "\[Proportional]", RowBox[List[FractionBox["1", SuperscriptBox["z", "2"]], RowBox[List["(", RowBox[List["1", "+", FractionBox["2", RowBox[List["3", SuperscriptBox["z", "2"]]]], "+", FractionBox["23", RowBox[List["45", SuperscriptBox["z", "4"]]]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox["z", "6"]], "]"]]]], ")"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <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> ∝ </mo> <mrow> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 2 </mn> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 23 </mn> <mrow> <mn> 45 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 6 </mn> </msup> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <power /> <apply> <arccoth /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 23 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 45 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 6 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["ArcCoth", "[", "z_", "]"]], "2"], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["1", "+", FractionBox["2", RowBox[List["3", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox["23", RowBox[List["45", " ", SuperscriptBox["z", "4"]]]], "+", RowBox[List["SeriesData", "[", RowBox[List["z", ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "6"]], "]"]]]], SuperscriptBox["z", "2"]]]]]]

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

2007-05-02