Hyperbolic sine: Representations through more general functions (formula 01.19.26.0074)

Sinh

$Mathematica Notation$

Elementary Functions

Sinh[z]

Representations through more general functions

Through other functions

Involving some elliptic-type functions

http://functions.wolfram.com/01.19.26.0074.01

Input Form

Sinh[z] == (-I) Sqrt[3/2] E^(z^2/6) WeierstrassSigma[Sqrt[2/3] I z, {3, 1}]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Sinh", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SqrtBox[FractionBox["3", "2"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], "6"]], RowBox[List["WeierstrassSigma", "[", RowBox[List[RowBox[List[SqrtBox[FractionBox["2", "3"]], " ", "\[ImaginaryI]", " ", "z"]], ",", RowBox[List["{", RowBox[List["3", ",", "1"]], "}"]]]], "]"]], " "]]]]]]

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> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mi> ⅈ </mi> </mrow> <mo> ⁢ </mo> <msqrt> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </msqrt> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 6 </mn> </mfrac> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mi> σ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mrow> <msqrt> <mfrac> <mn> 2 </mn> <mn> 3 </mn> </mfrac> </msqrt> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ; </mo> <mn> 3 </mn> </mrow> <mo> , </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Sigma]", "(", RowBox[List[RowBox[List[TagBox[RowBox[List[SqrtBox[FractionBox["2", "3"]], " ", "\[ImaginaryI]", " ", "z"]], Rule[Editable, True]], ";", TagBox["3", Rule[Editable, True]]]], ",", TagBox["1", Rule[Editable, True]]]], ")"]], InterpretTemplate[Function[WeierstrassSigma[Slot[1], List[Slot[2], Slot[3]]]]]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sinh /> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <cn type='rational'> 3 <sep /> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> WeierstrassSigma </ci> <apply> <times /> <apply> <power /> <cn type='rational'> 2 <sep /> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> <ci> z </ci> </apply> <list> <cn type='integer'> 3 </cn> <cn type='integer'> 1 </cn> </list> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sinh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SqrtBox[FractionBox["3", "2"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], "6"]], " ", RowBox[List["WeierstrassSigma", "[", RowBox[List[RowBox[List[SqrtBox[FractionBox["2", "3"]], " ", "\[ImaginaryI]", " ", "z"]], ",", RowBox[List["{", RowBox[List["3", ",", "1"]], "}"]]]], "]"]]]]]]]]

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

2001-10-29