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Sech






Mathematica Notation

Traditional Notation









Elementary Functions > Sech[z] > Representations through equivalent functions > With inverse function





http://functions.wolfram.com/01.24.27.0065.01









  


  










Input Form





ArcSech[Sech[z]] == Sqrt[z^2] (1 - (2 Pi I k)/z) /; (((2 k - 1) Pi < Im[z] < (2 k + 1) Pi || (Im[z] == (2 k - 1) Pi && Re[z] < 0) || (Im[z] == (2 k + 1) Pi && Re[z] > 0)) && Element[k, Integers]) || (z == (2 k - 1) Pi I && Element[-k, Integers] && -k >= 0) || (z == (2 k + 1) Pi I && Element[k, Integers] && k >= 0)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcSech", "[", RowBox[List["Sech", "[", "z", "]"]], "]"]], "\[Equal]", RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["2", "\[Pi]", " ", "\[ImaginaryI]", " ", "k"]], "z"]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "-", "1"]], ")"]], " ", "\[Pi]"]], "<", RowBox[List["Im", "[", "z", "]"]], "<", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], "\[Pi]"]]]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "-", "1"]], ")"]], " ", "\[Pi]"]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "<", "0"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], "\[Pi]"]]]], "\[And]", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]], ")"]]]], ")"]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["z", "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], " ", "-", "1"]], ")"]], "\[Pi]", " ", "\[ImaginaryI]"]]]], "\[And]", RowBox[List[RowBox[List["-", "k"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "k"]], "\[GreaterEqual]", "0"]]]], ")"]], "\[Or]", RowBox[List["(", RowBox[List[RowBox[List["z", "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], " ", "+", "1"]], ")"]], "\[Pi]", " ", "\[ImaginaryI]"]]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> sech </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> &lt; </mo> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &lt; </mo> <mn> 0 </mn> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mrow> <mi> Im </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &gt; </mo> <mn> 0 </mn> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <mo> &#8744; </mo> <mrow> <mrow> <mi> z </mi> <mo> &#10869; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; 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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSech", "[", RowBox[List["Sech", "[", "z_", "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox[SuperscriptBox["z", "2"]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]", " ", "k"]], "z"]]], ")"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], " ", "\[Pi]"]], "<", RowBox[List["Im", "[", "z", "]"]], "<", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", "\[Pi]"]]]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], " ", "\[Pi]"]]]], "&&", RowBox[List[RowBox[List["Re", "[", "z", "]"]], "<", "0"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Im", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", "\[Pi]"]]]], "&&", RowBox[List[RowBox[List["Re", "[", "z", "]"]], ">", "0"]]]], ")"]]]], ")"]], "&&", RowBox[List["k", "\[Element]", "Integers"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["z", "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "1"]], ")"]], " ", "\[Pi]", " ", "\[ImaginaryI]"]]]], "&&", RowBox[List[RowBox[List["-", "k"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "k"]], "\[GreaterEqual]", "0"]]]], ")"]], "||", RowBox[List["(", RowBox[List[RowBox[List["z", "\[Equal]", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], " ", "\[Pi]", " ", "\[ImaginaryI]"]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", "\[GreaterEqual]", "0"]]]], ")"]]]]]]]]]]










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





2003-08-21





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