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Cosh




Mathematica Notation

Traditional Notation









Elementary Functions > Cosh[z] > Series representations > Generalized power series > Expansions at z==Pi i/2 > For the function itself





http://functions.wolfram.com/01.20.06.0010.02









  


  








Input Form





Cosh[z] \[Proportional] I (z - (I Pi)/2) + (I/6) (z - (I Pi)/2)^3 + (I/120) (z - (I Pi)/2)^5 + \[Ellipsis] /; (z -> (I Pi)/2)








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Cosh", "[", "z", "]"]], "\[Proportional]", RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"]]], ")"]]]], "+", RowBox[List[FractionBox["\[ImaginaryI]", "6"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"]]], ")"]], "3"]]], "+", RowBox[List[FractionBox["\[ImaginaryI]", "120"], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"]]], ")"]], "5"]]], "+", "\[Ellipsis]"]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[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> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mi> &#8520; </mi> <mn> 6 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mi> &#8520; </mi> <mn> 120 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mn> 5 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <cosh /> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 120 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 5 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <apply> <times /> <imaginaryi /> <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["Cosh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"]]], ")"]]]], "+", RowBox[List[FractionBox["1", "6"], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"]]], ")"]], "3"]]], "+", RowBox[List[FractionBox["1", "120"], " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"]]], ")"]], "5"]]], "+", "\[Ellipsis]"]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "2"]]], ")"]]]]]]]]


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





2001-10-29