Inverse hyperbolic cotangent: Representations through equivalent functions (formula 01.28.27.2709)

ArcCoth

$Mathematica Notation$

Elementary Functions

ArcCoth[z]

Representations through equivalent functions

With related functions

Involving tanh^-1

Involving coth^-1(a (b z^c)^m)

Involving coth^-1(a (b z^c)^m) and tanh^-1(1/a b^-m z^{-m c})

http://functions.wolfram.com/01.28.27.2709.01

Input Form

ArcCoth[a (b z^c)^m] == ((b z^c)^m/(b^m z^(m c))) ArcTanh[1/a/(b^m z^(m c))] /; Element[2 m, Integers]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCoth", "[", RowBox[List["a", " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], ")"]], "m"]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", " ", SuperscriptBox["z", "c"]]], ")"]], "m"], " "]], RowBox[List[SuperscriptBox["b", "m"], " ", SuperscriptBox["z", RowBox[List["m", " ", "c"]]]]]], RowBox[List["ArcTanh", "[", RowBox[List[FractionBox["1", "a"], " ", SuperscriptBox["b", RowBox[List["-", "m"]]], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "m"]], " ", "c"]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["2", " ", "m"]], "\[Element]", "Integers"]]]]]]

MathML Form

<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> a </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mi> c </mi> </msup> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mtext> </mtext> </mrow> <mrow> <msup> <mi> b </mi> <mi> m </mi> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> c </mi> </mrow> </msup> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> a </mi> </mfrac> <mo> ⁢ </mo> <msup> <mi> b </mi> <mrow> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> z </mi> <mrow> <mrow> <mo> - </mo> <mi> m </mi> </mrow> <mo> ⁢ </mo> <mi> c </mi> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> m </mi> </mrow> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arccoth /> <apply> <times /> <ci> a </ci> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <ci> m </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <ci> c </ci> </apply> </apply> <ci> m </ci> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> b </ci> <ci> m </ci> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <ci> m </ci> <ci> c </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arctanh /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> </apply> <apply> <power /> <ci> z </ci> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> c </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> m </ci> </apply> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2003-09-04