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http://functions.wolfram.com/01.24.03.0035.01
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Sech[(3 Pi I)/7] == (12 2^(2/3) (7 - 21 I Sqrt[3])^(1/3))/
(2 2^(2/3) (7 - 21 I Sqrt[3])^(1/3) - 2 Sqrt[7] (I + Sqrt[3])
(7 - (I Sqrt[7])/2 - (3 Sqrt[21])/2)^(1/3) -
4 I Sqrt[7] (7 + (I Sqrt[7])/2 + (3 Sqrt[21])/2)^(1/3) -
2 (14 - I Sqrt[7] - 3 Sqrt[21])^(2/3) (14 + I Sqrt[7] + 3 Sqrt[21])^
(1/3) + (14 - I Sqrt[7] - 3 Sqrt[21])^(1/3)
(14 + I Sqrt[7] + 3 Sqrt[21])^(2/3) +
I Sqrt[3] (14 - I Sqrt[7] - 3 Sqrt[21])^(1/3)
(14 + I Sqrt[7] + 3 Sqrt[21])^(2/3))
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Cell[BoxData[RowBox[List[RowBox[List["Sech", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[ImaginaryI]"]], "7"], "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List["12", " ", 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[RowBox[List["2", " ", SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["7", "-", RowBox[List["21", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "-", RowBox[List["2", " ", SqrtBox["7"], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["7", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "2"], "-", FractionBox[RowBox[List["3", " ", SqrtBox["21"]]], "2"]]], ")"]], RowBox[List["1", "/", "3"]]]]], "-", RowBox[List["4", " ", "\[ImaginaryI]", " ", SqrtBox["7"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["7", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "2"], "+", FractionBox[RowBox[List["3", " ", SqrtBox["21"]]], "2"]]], ")"]], RowBox[List["1", "/", "3"]]]]], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["14", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "-", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["14", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "+", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["14", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "-", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["1", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["14", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "+", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["2", "/", "3"]]]]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["14", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "-", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["1", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["14", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "+", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["2", "/", "3"]]]]]]], ")"]]]]]]]]
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<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> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> <mn> 7 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 12 </mn> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> ⁢ </mo> <mroot> <mrow> <mn> 7 </mn> <mo> - </mo> <mrow> <mn> 21 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> ) </mo> </mrow> <mo> / </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 4 </mn> </mrow> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 7 </mn> </msqrt> <mo> ⁢ </mo> <mroot> <mrow> <mn> 7 </mn> <mo> + </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 7 </mn> </msqrt> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msqrt> <mn> 21 </mn> </msqrt> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mn> 7 </mn> </msqrt> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> ⅈ </mi> <mo> + </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mroot> <mrow> <mn> 7 </mn> <mo> - </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 7 </mn> </msqrt> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msqrt> <mn> 21 </mn> </msqrt> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> ⁢ </mo> <mroot> <mrow> <mn> 7 </mn> <mo> - </mo> <mrow> <mn> 21 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 3 </mn> </msqrt> </mrow> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 14 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 7 </mn> </msqrt> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msqrt> <mn> 21 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> ⁢ </mo> <mroot> <mrow> <mn> 14 </mn> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 7 </mn> </msqrt> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msqrt> <mn> 21 </mn> </msqrt> </mrow> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> + </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 14 </mn> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 7 </mn> </msqrt> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msqrt> <mn> 21 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> ⁢ </mo> <mroot> <mrow> <mn> 14 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 7 </mn> </msqrt> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msqrt> <mn> 21 </mn> </msqrt> </mrow> </mrow> <mn> 3 </mn> </mroot> </mrow> <mo> + </mo> <mrow> <msqrt> <mn> 3 </mn> </msqrt> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 14 </mn> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 7 </mn> </msqrt> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msqrt> <mn> 21 </mn> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> / </mo> <mn> 3 </mn> </mrow> </msup> <mo> ⁢ </mo> <mroot> <mrow> <mn> 14 </mn> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <msqrt> <mn> 7 </mn> </msqrt> </mrow> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> ⁢ </mo> <msqrt> <mn> 21 </mn> </msqrt> </mrow> </mrow> <mn> 3 </mn> </mroot> <mo> ⁢ </mo> <mi> ⅈ </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sech /> <apply> <times /> <cn type='integer'> 3 </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'> 12 </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> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -4 </cn> <imaginaryi /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 7 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 21 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <imaginaryi /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 7 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 7 </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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 21 </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'> 1 <sep /> 3 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </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> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 14 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 21 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 14 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 21 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 14 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 21 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 14 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 21 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 14 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 21 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 2 <sep /> 3 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 14 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 7 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 21 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 3 </cn> </apply> <imaginaryi /> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sech", "[", FractionBox[RowBox[List["3", " ", "\[Pi]", " ", "\[ImaginaryI]"]], "7"], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["12", " ", 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["2", " ", SuperscriptBox["2", RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["7", "-", RowBox[List["21", " ", "\[ImaginaryI]", " ", SqrtBox["3"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "-", RowBox[List["2", " ", SqrtBox["7"], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", "+", SqrtBox["3"]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["7", "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "2"], "-", FractionBox[RowBox[List["3", " ", SqrtBox["21"]]], "2"]]], ")"]], RowBox[List["1", "/", "3"]]]]], "-", RowBox[List["4", " ", "\[ImaginaryI]", " ", SqrtBox["7"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["7", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "2"], "+", FractionBox[RowBox[List["3", " ", SqrtBox["21"]]], "2"]]], ")"]], RowBox[List["1", "/", "3"]]]]], "-", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["14", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "-", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["2", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["14", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "+", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["1", "/", "3"]]]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["14", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "-", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["1", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["14", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "+", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["2", "/", "3"]]]]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["3"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["14", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "-", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["1", "/", "3"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["14", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["7"]]], "+", RowBox[List["3", " ", SqrtBox["21"]]]]], ")"]], RowBox[List["2", "/", "3"]]]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
