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Sech




Mathematica Notation

Traditional Notation









Elementary Functions > Sech[z] > Specific values > Values at fixed points





http://functions.wolfram.com/01.24.03.0111.01









  


  








Input Form





Sech[(10 Pi I)/7] == -(6 2^(2/3) (7 - 21 I Sqrt[3])^(1/3))/ (7 + 7 I Sqrt[3] + ((7/2) (1 - 3 I Sqrt[3]))^(2/3) - I Sqrt[3] (7/2 - (21 I Sqrt[3])/2)^(2/3) + 2^(2/3) (7 - 21 I Sqrt[3])^(1/3))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Sech", "[", FractionBox[RowBox[List["10", " ", "\[Pi]", " ", "\[ImaginaryI]"]], "7"], "]"]], "\[Equal]", RowBox[List[RowBox[List["-", RowBox[List["(", RowBox[List["6", " ", SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["7", "-", RowBox[List["21", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], ")"]]]], "/", RowBox[List["(", RowBox[List["7", "+", RowBox[List["7", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["7", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]]]], ")"]], RowBox[List["2", "/", "3"]]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["7", "2"], "-", FractionBox[RowBox[List["21", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]], "2"]]], ")"]], RowBox[List["2", "/", "3"]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["7", "-", RowBox[List["21", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]]]], ")"]]]]]]]]








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> <mfrac> <mrow> <mn> 10 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 7 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mo> ( </mo> <mrow> <mn> 6 </mn> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <mroot> <mrow> <mn> 7 </mn> <mo> - </mo> <mrow> <mn> 21 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mn> 2 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> &#8290; </mo> <mroot> <mrow> <mn> 7 </mn> <mo> - </mo> <mrow> <mn> 21 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mn> 3 </mn> </msqrt> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 7 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 21 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> </mrow> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 7 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msqrt> <mn> 3 </mn> </msqrt> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mo> + </mo> <mn> 7 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sech /> <apply> <times /> <cn type='integer'> 10 </cn> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 7 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 21 </cn> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 7 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 21 </cn> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='rational'> 7 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 21 </cn> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 2 <sep /> 3 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='rational'> 7 <sep /> 2 </cn> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <imaginaryi /> </apply> <cn type='integer'> 7 </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", "[", FractionBox[RowBox[List["10", " ", "\[Pi]", " ", "\[ImaginaryI]"]], "7"], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["6", " ", SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["7", "-", RowBox[List["21", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], RowBox[List["7", "+", RowBox[List["7", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]], "+", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["7", "2"], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["3", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]]]], ")"]], RowBox[List["2", "/", "3"]]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[FractionBox["7", "2"], "-", FractionBox[RowBox[List["21", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]], "2"]]], ")"]], RowBox[List["2", "/", "3"]]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["7", "-", RowBox[List["21", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]]]]]]]]]]]


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





2001-10-29