Inverse hyperbolic cotangent: Representations through equivalent functions (formula 01.28.27.0831)

ArcCoth

$Mathematica Notation$

Elementary Functions

Representations through equivalent functions

With related functions

Involving tan^-1

Involving coth^-1((-z)^1/2)

Involving coth^-1((-z)^1/2) and tan^-1(1/z^1/2)

http://functions.wolfram.com/01.28.27.0831.01

Input Form

ArcCoth[Sqrt[-z]] == (-(Sqrt[-z]/Sqrt[z])) ArcTan[1/Sqrt[z]]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["ArcCoth", "[", SqrtBox[RowBox[List["-", "z"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SqrtBox[RowBox[List["-", "z"]]], SqrtBox["z"]]]], RowBox[List["ArcTan", "[", FractionBox["1", SqrtBox["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> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <msqrt> <mi> z </mi> </msqrt> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <msqrt> <mi> z </mi> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccoth /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <arctan /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

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

2003-09-04