Inverse hyperbolic cotangent: Representations through equivalent functions (formula 01.28.27.1058)

ArcCoth

$Mathematica Notation$

Elementary Functions

ArcCoth[z]

Representations through equivalent functions

With related functions

Involving cot^-1

Involving coth^-1(1+z²/2z)

Involving coth^-1(1+z²/2z) and cot^-1(i z)

http://functions.wolfram.com/01.28.27.1058.01

Input Form

ArcCoth[(1 + z^2)/(2 z)] == z Sqrt[-z^(-2)] Pi + 2 I ArcCot[I z] /; Abs[z] < 1

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCoth", "[", FractionBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]], RowBox[List["2", "z"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["z", SqrtBox[RowBox[List["-", SuperscriptBox["z", RowBox[List["-", "2"]]]]]], "\[Pi]"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcCot", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "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> <mrow> <msup> <mi> coth </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> cot </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <mo> < </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arccoth /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <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> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <arccot /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <lt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCoth", "[", FractionBox[RowBox[List["1", "+", SuperscriptBox["z_", "2"]]], RowBox[List["2", " ", "z_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["z", " ", SqrtBox[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]], " ", "\[Pi]"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["ArcCot", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]]]]]]]]

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

2003-09-04