Hyperbolic secant: Representations through equivalent functions (formula 01.24.27.0051)

Sech

$Mathematica Notation$

Elementary Functions

Sech[z]

Representations through equivalent functions

With related functions

Involving csch

http://functions.wolfram.com/01.24.27.0051.01

Input Form

Sech[z] == Csch[z]/Sqrt[Csch[z]^2 + 1] /; Re[z] > 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Sech", "[", "z", "]"]], "\[Equal]", FractionBox[RowBox[List[" ", RowBox[List["Csch", "[", "z", "]"]]]], SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["Csch", "[", "z", "]"]], "2"], "+", "1"]]]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]

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> <mfrac> <mrow> <mtext> </mtext> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mrow> <msup> <mi> csch </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> > </mo> <mn> 0 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sech /> <ci> z </ci> </apply> <apply> <times /> <apply> <csch /> <ci> z </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <csch /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sech", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["Csch", "[", "z", "]"]], SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["Csch", "[", "z", "]"]], "2"], "+", "1"]]]], "/;", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]]]]

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

2001-10-29