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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 sec-1





http://functions.wolfram.com/01.24.16.0070.01









  


  








Input Form





Sech[a ArcSec[z]] == (2 E^(a (Pi/2)) (Sqrt[1 - 1/z^2] + I/z)^(a I))/ (1 + E^(a Pi) (Sqrt[1 - 1/z^2] + I/z)^(2 a I))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Sech", "[", RowBox[List["a", " ", RowBox[List["ArcSec", "[", "z", "]"]]]], "]"]], "\[Equal]", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", RowBox[List["\[Pi]", "/", "2"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["z", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "z"]]], ")"]], RowBox[List["a", " ", "\[ImaginaryI]"]]]]], RowBox[List["1", "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", "\[Pi]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["z", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "z"]]], ")"]], RowBox[List["2", " ", "a", " ", "\[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> a </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> sec </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> <mi> &#8519; </mi> <mfrac> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> &#8520; </mi> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </msup> </mrow> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mn> 1 </mn> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> + </mo> <mfrac> <mi> &#8520; </mi> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </msup> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sech /> <apply> <times /> <ci> a </ci> <apply> <arcsec /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <exponentiale /> <apply> <times /> <ci> a </ci> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <ci> a </ci> <imaginaryi /> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> a </ci> <pi /> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <imaginaryi /> </apply> </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["a_", " ", RowBox[List["ArcSec", "[", "z_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["2", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["a", " ", "\[Pi]"]], "2"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["z", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "z"]]], ")"]], RowBox[List["a", " ", "\[ImaginaryI]"]]]]], RowBox[List["1", "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", "\[Pi]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["z", "2"]]]]], "+", FractionBox["\[ImaginaryI]", "z"]]], ")"]], RowBox[List["2", " ", "a", " ", "\[ImaginaryI]"]]]]]]]]]]]]


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





2007-05-02