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http://functions.wolfram.com/01.24.27.0034.01
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Sech[z] == (1/Sqrt[1 + Sinh[z]^2]) (-1)^Floor[1/2 + Im[z]/Pi]
(1 - (1 + (-1)^(Floor[-(1/2) - Im[z]/Pi] + Floor[1/2 + Im[z]/Pi]))
UnitStep[Re[z]])
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Cell[BoxData[RowBox[List[RowBox[List["Sech", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox["1", SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "2"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"]]], "]"]]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"]]], "]"]]]]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]]]], ")"]]]]]]]]
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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> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <mrow> <msup> <mi> sinh </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> ⌊ </mo> <mrow> <mfrac> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> π </mi> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⌋ </mo> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> ⌊ </mo> <mrow> <mfrac> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> π </mi> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mrow> <mo> ⌊ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mi> π </mi> </mfrac> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> ⌋ </mo> </mrow> </mrow> </msup> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> θ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> Re </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sech /> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <sinh /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <plus /> <apply> <times /> <apply> <imaginary /> <ci> z </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <apply> <floor /> <apply> <plus /> <apply> <times /> <apply> <imaginary /> <ci> z </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <floor /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <imaginary /> <ci> z </ci> </apply> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> θ </ci> <apply> <real /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sech", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"]]], "]"]]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["Floor", "[", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"]]], "]"]], "+", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["Im", "[", "z", "]"]], "\[Pi]"]]], "]"]]]]]]], ")"]], " ", RowBox[List["UnitStep", "[", RowBox[List["Re", "[", "z", "]"]], "]"]]]]]], ")"]]]], SqrtBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["Sinh", "[", "z", "]"]], "2"]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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