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Cosh




Mathematica Notation

Traditional Notation









Elementary Functions > Cosh[z] > Transformations > Transformations and argument simplifications > Argument involving inverse trigonometric and hyperbolic functions > Involving sech-1





http://functions.wolfram.com/01.20.16.0151.01









  


  








Input Form





Cosh[I ArcSech[z]] == ((1/2) (1 + (Sqrt[-1 + 1/z] Sqrt[1 + 1/z] + 1/z)^(2 I)))/ (Sqrt[-1 + 1/z] Sqrt[1 + 1/z] + 1/z)^I








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSech", "[", "z", "]"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", FractionBox["1", "z"]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "z"]]]]]], "+", FractionBox["1", "z"]]], ")"]], RowBox[List["2", " ", "\[ImaginaryI]"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", FractionBox["1", "z"]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "z"]]]]]], "+", FractionBox["1", "z"]]], ")"]], RowBox[List["-", "\[ImaginaryI]"]]]]]]]]]








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> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> </msqrt> </mrow> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> &#8520; </mi> </mrow> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <cosh /> <apply> <times /> <imaginaryi /> <apply> <arcsech /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Cosh", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcSech", "[", "z_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", FractionBox["1", "z"]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "z"]]]]]], "+", FractionBox["1", "z"]]], ")"]], RowBox[List["2", " ", "\[ImaginaryI]"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", FractionBox["1", "z"]]]], " ", SqrtBox[RowBox[List["1", "+", FractionBox["1", "z"]]]]]], "+", FractionBox["1", "z"]]], ")"]], RowBox[List["-", "\[ImaginaryI]"]]]]]]]]]


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





2007-05-02