Hyperbolic sine: Representations through more general functions (formula 01.19.26.0075)

Sinh

$Mathematica Notation$

Elementary Functions

Sinh[z]

Representations through more general functions

Through other functions

Involving some hypergeometric-type functions

http://functions.wolfram.com/01.19.26.0075.01

Input Form

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

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Sinh", "[", RowBox[List["\[Pi]", " ", "z"]], "]"]], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[ImaginaryI]", " ", "z"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List["1", "-", 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> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <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> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sinh /> <apply> <times /> <pi /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> </apply> <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> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

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

2001-10-29