Inverse hyperbolic cotangent: Representations through equivalent functions (formula 01.28.27.2797)

ArcCoth

$Mathematica Notation$

Elementary Functions

Representations through equivalent functions

With related functions

Involving tanh^-1

Involving coth^-1(z/(1-z²)^1/2+a)

Involving coth^-1(z/(1-z²)^1/2-1) and tanh^-1(z)

http://functions.wolfram.com/01.28.27.2797.01

Input Form

ArcCoth[z/(Sqrt[1 - z^2] - 1)] == (-(1/2)) ArcTanh[z]

Standard Form

Cell[BoxData[RowBox[List[" ", RowBox[List[RowBox[List["ArcCoth", "[", FractionBox["z", RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], "-", "1"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], RowBox[List["ArcTanh", "[", "z", "]"]]]]]]]]]]

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> <mi> coth </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> z </mi> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccoth /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arctanh /> <ci> z </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCoth", "[", FractionBox["z_", RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["z_", "2"]]]], "-", "1"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["ArcTanh", "[", "z", "]"]], "2"]]]]]]]

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

2003-09-04