Hyperbolic secant: Representations through equivalent functions (formula 01.24.27.0066)

Sech

$Mathematica Notation$

Elementary Functions

Sech[z]

Representations through equivalent functions

With inverse function

http://functions.wolfram.com/01.24.27.0066.01

Input Form

ArcSech[Sech[z]] == Piecewise[ {{(-(-1)^Floor[(Pi - I z)/Pi]) (z - Pi I Floor[(Pi - I z)/Pi] + (Pi I)/2) + (Pi I)/2, Re[z] == 0}, {Sqrt[z^2] (1 - ((2 Pi I)/z) Floor[(Im[z] - Pi)/(2 Pi)]), Element[(Im[z] + Pi)/(2 Pi), Integers] && Re[z] > 0}, {Sqrt[z^2] (1 - ((2 Pi I)/z) Floor[(Im[z] + Pi)/(2 Pi)]), True}}]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["Sech", "[", "z", "]"]], "]"]], "\[Equal]", RowBox[List["Piecewise", "[", RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "\[Pi]"], "]"]]]]], RowBox[List["(", RowBox[List["z", "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "\[Pi]"], "]"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]], ",", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[Equal]", "0"]]]], "}"]], ",", RowBox[List["{", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox[RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]"]], "z"], RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Im", "[", "z", "]"]], "-", "\[Pi]"]], RowBox[List["2", "\[Pi]"]]], "]"]]]]]], ")"]]]], ",", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Im", "[", "z", "]"]], "+", "\[Pi]"]], RowBox[List["2", "\[Pi]"]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]]], "}"]], ",", "\[IndentingNewLine]", RowBox[List["{", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[FractionBox[RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]"]], "z"], RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Im", "[", "z", "]"]], "+", "\[Pi]"]], RowBox[List["2", "\[Pi]"]]], "]"]]]]]], ")"]]]], ",", "True"]], "}"]]]], "}"]], "]"]]]]]]

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> 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> <mtable> <mtr> <mtd> <mrow> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> π </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mi> π </mi> </mfrac> <mo> ⌋ </mo> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mi> π </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mi> π </mi> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mtd> <mtd> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo>  </mo> <mn> 0 </mn> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mi> z </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mtd> <mtd> <mrow> <mrow> <mfrac> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ∈ </mo> <semantics> <mi> ℤ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalZ]", Function[List[], Integers]] </annotation> </semantics> </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> </mtd> </mtr> <mtr> <mtd> <mrow> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mi> z </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mtd> <mtd> <mi> True </mi> </mtd> </mtr> </mtable> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arcsech /> <apply> <sech /> <ci> z </ci> </apply> </apply> <apply> <ci> Piecewise </ci> <list> <list> <apply> <plus /> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> <apply> <floor /> <apply> <times /> <apply> <plus /> <pi /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </list> <list> <apply> <times /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <imaginary /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <pi /> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <apply> <times /> <apply> <plus /> <apply> <imaginary /> <ci> z </ci> </apply> <pi /> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <integers /> </apply> <apply> <gt /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </list> <list> <apply> <times /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <imaginary /> <ci> z </ci> </apply> <pi /> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <true /> </list> </list> </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["\[Piecewise]", GridBox[List[List[RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "\[Pi]"], "]"]]]]], " ", RowBox[List["(", RowBox[List["z", "-", RowBox[List["\[Pi]", " ", "\[ImaginaryI]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "\[Pi]"], "]"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], ")"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], "2"]]], RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[Equal]", "0"]]], List[RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Im", "[", "z", "]"]], "-", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "z"]]], ")"]]]], RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["Im", "[", "z", "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]]], List[RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]], " ", RowBox[List["Floor", "[", FractionBox[RowBox[List[RowBox[List["Im", "[", "z", "]"]], "+", "\[Pi]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "z"]]], ")"]]]], "True"]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]]]]]]

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

2007-05-02