Cosecant: Representations through equivalent functions (formula 01.10.27.0068)

Csc

$Mathematica Notation$

Elementary Functions

Csc[z]

Representations through equivalent functions

With related functions

Involving sech

http://functions.wolfram.com/01.10.27.0068.01

Input Form

Csc[z] == (-z) Sqrt[-(1/z^2)] (Sech[I z]/Sqrt[1 - Sech[I z]^2]) /; Im[z] != 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Csc", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", "z"]], SqrtBox[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]], FractionBox[RowBox[List["Sech", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]], SqrtBox[RowBox[List["1", "-", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]], "2"]]]]]]]]], "/;", RowBox[List[RowBox[List["Im", "[", "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> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mfrac> <mrow> <mi> sech </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mi> sech </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msqrt> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> Im </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> <csc /> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <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> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <apply> <sech /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <sech /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <neq /> <apply> <imaginary /> <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["Csc", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List["z", " ", SqrtBox[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]], " ", RowBox[List["Sech", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox[RowBox[List["Sech", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]], "2"]]]]]]], "/;", RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[NotEqual]", "0"]]]]]]]]

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

2001-10-29