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Sinh




Mathematica Notation

Traditional Notation









Elementary Functions > Sinh[z] > Representations through equivalent functions > With related functions > Involving cos





http://functions.wolfram.com/01.19.27.0009.01









  


  








Input Form





Sinh[z] == (Sqrt[z^2]/z) Sqrt[(Cos[2 I z] - 1)/2] /; Abs[Im[z]] < Pi/2








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Sinh", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " "]], "z"], SqrtBox[FractionBox[RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", "\[ImaginaryI]", " ", "z"]], "]"]], "-", "1"]], "2"]]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "<", FractionBox["\[Pi]", "2"]]]]]]]








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> sinh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mtext> </mtext> </mrow> <mi> z </mi> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> </msqrt> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &lt; </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sinh /> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <cos /> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <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> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sinh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", SqrtBox[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List["2", " ", "\[ImaginaryI]", " ", "z"]], "]"]], "-", "1"]], ")"]]]]]]], "z"], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Im", "[", "z", "]"]], "]"]], "<", FractionBox["\[Pi]", "2"]]]]]]]]]


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





2001-10-29