Hyperbolic secant: Representations through equivalent functions (formula 01.24.27.0015)

Sech

$Mathematica Notation$

Elementary Functions

Sech[z]

Representations through equivalent functions

With related functions

Involving tan

http://functions.wolfram.com/01.24.27.0015.01

Input Form

Sech[z] == Sqrt[1 + Tan[I z]^2] /; Abs[Im[z]] < Pi/2

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Sech", "[", "z", "]"]], "\[Equal]", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]], "2"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "<", 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> <mi> sech </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <msup> <mi> tan </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </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> <sech /> <ci> z </ci> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <tan /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <lt /> <apply> <abs /> <apply> <imaginary /> <ci> z </ci> </apply> </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["Sech", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Tan", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]], "2"]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "<", FractionBox["\[Pi]", "2"]]]]]]]]]

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

2001-10-29