Inverse hyperbolic cotangent: Representations through equivalent functions (formula 01.28.27.2643)

ArcCoth

$Mathematica Notation$

Elementary Functions

ArcCoth[z]

Representations through equivalent functions

With related functions

Involving cosh^-1

Involving coth^-1(z/(1+z²)^1/2+a)

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

http://functions.wolfram.com/01.28.27.2643.01

Input Form

ArcCoth[z/(Sqrt[1 + z^2] + 1)] == -((3 Pi I)/4) + (1/2) ArcCosh[I z] /; 0 <= Arg[z] <= Pi/2

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[RowBox[List["3", "\[Pi]", " ", "\[ImaginaryI]"]], "4"]]], "+", RowBox[List[FractionBox["1", "2"], RowBox[List["ArcCosh", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]]]]]], "/;", RowBox[List["0", "\[LessEqual]", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]

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> <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> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mn> 4 </mn> </mfrac> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> cosh </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> <mn> 0 </mn> <mo> ≤ </mo> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ≤ </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arccoth /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </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> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <arccosh /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <leq /> <cn type='integer'> 0 </cn> <apply> <arg /> <ci> z </ci> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </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[RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["3", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["ArcCosh", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]]]], "/;", RowBox[List["0", "\[LessEqual]", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]]]]]]

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

2003-09-04