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Sech




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.24.16.0073.01









  


  








Input Form





Sech[I ArcCosh[z]] == (2 (z + Sqrt[-1 + z] Sqrt[1 + z])^I)/ (1 + (z + Sqrt[-1 + z] Sqrt[1 + z])^(2 I))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Sech", "[", RowBox[List["\[ImaginaryI]", " ", RowBox[List["ArcCosh", "[", "z", "]"]]]], "]"]], "\[Equal]", FractionBox[RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]]], " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]], "\[ImaginaryI]"]]], RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]]], " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]], RowBox[List["2", " ", "\[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> sech </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> &#8520; </mi> </msup> </mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <msqrt> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sech /> <apply> <times /> <imaginaryi /> <apply> <arccosh /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <imaginaryi /> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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


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





2007-05-02