Hyperbolic secant: Representations through equivalent functions (formula 01.24.27.0064)

Sech

$Mathematica Notation$

Elementary Functions

Sech[z]

Representations through equivalent functions

With inverse function

http://functions.wolfram.com/01.24.27.0064.01

Input Form

ArcSech[Sech[z]] == -z /; (Re[z] < 0 && Inequality[-Pi, LessEqual, Im[z], Less, Pi]) || (Re[z] == 0 && -Pi <= Im[z] <= 0)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["Sech", "[", "z", "]"]], "]"]], "\[Equal]", RowBox[List["-", "z"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], "<", "0"]], "\[And]", RowBox[List[RowBox[List["-", "\[Pi]"]], "\[LessEqual]", RowBox[List["Im", "[", "z", "]"]], "<", "\[Pi]"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[Equal]", "0"]], "\[And]", RowBox[List[RowBox[List["-", "\[Pi]"]], "\[LessEqual]", RowBox[List["Im", "[", "z", "]"]], "\[LessEqual]", "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> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> sech </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mi> π </mi> </mrow> <mo> ≤ </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mi> π </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ∨ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <mrow> <mo> - </mo> <mi> π </mi> </mrow> <mo> ≤ </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ≤ </mo> <mn> 0 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arcsech /> <apply> <sech /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <or /> <apply> <and /> <apply> <lt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <ci> Inequality </ci> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <leq /> <apply> <imaginary /> <ci> z </ci> </apply> <lt /> <pi /> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <leq /> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> <apply> <imaginary /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSech", "[", RowBox[List["Sech", "[", "z_", "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", "z"]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], "<", "0"]], "&&", RowBox[List[RowBox[List["-", "\[Pi]"]], "\[LessEqual]", RowBox[List["Im", "[", "z", "]"]], "<", "\[Pi]"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List["-", "\[Pi]"]], "\[LessEqual]", RowBox[List["Im", "[", "z", "]"]], "\[LessEqual]", "0"]]]], ")"]]]]]]]]]]

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

2003-08-21