Hyperbolic cosecant: Representations through more general functions (formula 01.23.26.0025)

Csch

$Mathematica Notation$

Elementary Functions

Csch[z]

Representations through more general functions

Through other functions

Involving some hypergeometric-type functions

http://functions.wolfram.com/01.23.26.0025.01

Input Form

Csch[Pi z] == I ((Gamma[1 - I z] Gamma[I z])/Pi)

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Csch", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]], "\[Equal]", RowBox[List["\[ImaginaryI]", " ", FractionBox[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]], "\[Pi]"]]]]]]]

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> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mi> π </mi> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <csch /> <apply> <times /> <pi /> <ci> z </ci> </apply> </apply> <apply> <times /> <imaginaryi /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Csch", "[", RowBox[List["\[Pi]", " ", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[RowBox[List["Gamma", "[", RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]]]], ")"]]]], "\[Pi]"]]]]]

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

2001-10-29