Hyperbolic tangent: Representations through equivalent functions (formula 01.21.27.0051)

Tanh

$Mathematica Notation$

Elementary Functions

Tanh[z]

Representations through equivalent functions

With related functions

Involving cosh

http://functions.wolfram.com/01.21.27.0051.01

Input Form

Tanh[z] == (I Sqrt[1 - Cosh[z]^2])/Cosh[z] /; 0 < Im[z] < Pi

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Tanh", "[", "z", "]"]], "\[Equal]", FractionBox[RowBox[List["\[ImaginaryI]", SqrtBox[RowBox[List["1", "-", SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "2"]]]]]], RowBox[List["Cosh", "[", "z", "]"]]]]], "/;", RowBox[List["0", "<", RowBox[List["Im", "[", "z", "]"]], "<", "\[Pi]"]]]]]]

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> <mi> tanh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mi> cosh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mn> 0 </mn> <mo> < </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mi> π </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <tanh /> <ci> z </ci> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <cosh /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <cosh /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <apply> <imaginary /> <ci> z </ci> </apply> <pi /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Tanh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox[RowBox[List["Cosh", "[", "z", "]"]], "2"]]]]]], RowBox[List["Cosh", "[", "z", "]"]]], "/;", RowBox[List["0", "<", RowBox[List["Im", "[", "z", "]"]], "<", "\[Pi]"]]]]]]]]

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

2001-10-29