Hyperbolic secant: Representations through more general functions (formula 01.24.26.0006)

Sech

$Mathematica Notation$

Elementary Functions

Sech[z]

Representations through more general functions

Through other functions

Involving Bessel functions

http://functions.wolfram.com/01.24.26.0006.01

Input Form

Sech[z] == -(Sqrt[2/Pi]/(Sqrt[I z] BesselY[1/2, I z]))

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Sech", "[", "z", "]"]], "\[Equal]", RowBox[List["-", FractionBox[SqrtBox[FractionBox["2", "\[Pi]"]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", RowBox[List["BesselY", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mi> sech </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <msqrt> <mfrac> <mn> 2 </mn> <mi> π </mi> </mfrac> </msqrt> </mrow> <mo> ⁢ </mo> <mfrac> <mn> 1 </mn> <mrow> <msqrt> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msqrt> <mo> ⁢ </mo> <mrow> <msub> <mi> Y </mi> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </msub> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sech /> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ci> BesselY </ci> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <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["-", FractionBox[SqrtBox[FractionBox["2", "\[Pi]"]], RowBox[List[SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", RowBox[List["BesselY", "[", RowBox[List[FractionBox["1", "2"], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]]]]]]]]

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

2001-10-29