Inverse hyperbolic tangent: Representations through equivalent functions (formula 01.27.27.0836)

ArcTanh

$Mathematica Notation$

Elementary Functions

Representations through equivalent functions

With related functions

Involving tan^-1

Involving tanh^-1((z₂)^1/2)

Involving tanh^-1((z₂)^1/2) and tan^-1(i z)

http://functions.wolfram.com/01.27.27.0836.01

Input Form

ArcTanh[Sqrt[z^2]] == (-(Sqrt[z^2]/z)) I ArcTan[I z]

Standard Form

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

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcTanh", "[", SqrtBox[SuperscriptBox["z_", "2"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", "\[ImaginaryI]", " ", RowBox[List["ArcTan", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]], "z"]]]]]]]

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

2003-08-21