Inverse hyperbolic cotangent: Representations through equivalent functions (formula 01.28.27.1158)

ArcCoth

$Mathematica Notation$

Elementary Functions

Representations through equivalent functions

With related functions

Involving cot^-1

Involving coth^-1(1/(z²-1)^1/2+c z)

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

http://functions.wolfram.com/01.28.27.1158.01

Input Form

ArcCoth[1/(Sqrt[z^2 - 1] - z)] == (Pi Sqrt[-1 - z] (z - Sqrt[z^2]))/(4 Sqrt[1 + z] z) - (I/2) ArcCot[I z]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["ArcCoth", "[", FractionBox["1", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]], "-", "z"]]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["\[Pi]", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", "z"]]], " ", RowBox[List["(", RowBox[List["z", "-", SqrtBox[SuperscriptBox["z", "2"]]]], ")"]]]], RowBox[List["4", SqrtBox[RowBox[List["1", "+", "z"]]], " ", "z"]]], "-", RowBox[List[FractionBox["\[ImaginaryI]", "2"], RowBox[List["ArcCot", "[", RowBox[List["\[ImaginaryI]", " ", "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> <mn> 1 </mn> <mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mfrac> <mo> - </mo> <mrow> <mfrac> <mi> ⅈ </mi> <mn> 2 </mn> </mfrac> <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> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccoth /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arccot /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCoth", "[", FractionBox["1", RowBox[List[SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "-", "1"]]], "-", "z_"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", "z"]]], " ", RowBox[List["(", RowBox[List["z", "-", SqrtBox[SuperscriptBox["z", "2"]]]], ")"]]]], RowBox[List["4", " ", SqrtBox[RowBox[List["1", "+", "z"]]], " ", "z"]]], "-", RowBox[List[FractionBox["1", "2"], " ", "\[ImaginaryI]", " ", RowBox[List["ArcCot", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]]]]]]]]

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

2003-09-04