Hyperbolic cosecant: Representations through equivalent functions (formula 01.23.27.0053)

Csch

$Mathematica Notation$

Elementary Functions

Csch[z]

Representations through equivalent functions

With related functions

Involving coth

http://functions.wolfram.com/01.23.27.0053.01

Input Form

Csch[z] == Sqrt[1/z^2] z Sqrt[Coth[z]^2 - 1] /; Abs[Im[z]] < Pi/2 && Re[z] != 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Csch", "[", "z", "]"]], "\[Equal]", RowBox[List[SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]], "z", " ", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["Coth", "[", "z", "]"]], "2"], "-", "1"]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "<", FractionBox["\[Pi]", "2"]]], "\[And]", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[NotEqual]", "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> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msqrt> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </msqrt> <mo> ⁢ </mo> <mi> z </mi> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <msup> <mi> coth </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> <mo> /; </mo> <mrow> <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> <mo> ∧ </mo> <mrow> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ≠ </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <csch /> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <coth /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <and /> <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> <neq /> <apply> <real /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Csch", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]], " ", "z", " ", SqrtBox[RowBox[List[SuperscriptBox[RowBox[List["Coth", "[", "z", "]"]], "2"], "-", "1"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "<", FractionBox["\[Pi]", "2"]]], "&&", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "\[NotEqual]", "0"]]]]]]]]]]

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

2001-10-29