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Sech




Mathematica Notation

Traditional Notation









Elementary Functions > Sech[z] > Representations through more general functions > Through hypergeometric functions





http://functions.wolfram.com/01.24.26.0029.01









  


  








Input Form





Sech[z] == ((4 Pi)/(Pi^2 + 4 z^2)) HypergeometricPFQ[ {1, 3/2, 1/2 - (I z)/Pi, 1/2 + (I z)/Pi}, {1/2, 3/2 - (I z)/Pi, 3/2 + (I z)/Pi}, -1]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Sech", "[", "z", "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List["4", " ", "\[Pi]", " "]], RowBox[List[SuperscriptBox["\[Pi]", "2"], "+", RowBox[List["4", " ", SuperscriptBox["z", "2"]]]]]], RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox["3", "2"], ",", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Pi]"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Pi]"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Pi]"]]], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Pi]"]]]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]]]]]]]]








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> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mrow> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> <mo> &#8290; </mo> <semantics> <mrow> <mrow> <msub> <mo> &#8202; </mo> <mn> 4 </mn> </msub> <msub> <mi> F </mi> <mn> 3 </mn> </msub> </mrow> <mo> &#8289; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 1 </mn> <mo> , </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mi> &#960; </mi> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mi> &#960; </mi> </mfrac> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mi> &#960; </mi> </mfrac> </mrow> <mo> , </mo> <mrow> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mi> &#960; </mi> </mfrac> <mo> + </mo> <mfrac> <mn> 3 </mn> <mn> 2 </mn> </mfrac> </mrow> </mrow> <mo> ; </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox[&quot;\[InvisiblePrefixScriptBase]&quot;, &quot;4&quot;], SubscriptBox[&quot;F&quot;, &quot;3&quot;]]], &quot;\[InvisibleApplication]&quot;, RowBox[List[&quot;(&quot;, RowBox[List[TagBox[TagBox[RowBox[List[TagBox[&quot;1&quot;, HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[FractionBox[&quot;3&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;1&quot;, &quot;2&quot;], &quot;-&quot;, FractionBox[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;z&quot;]], &quot;\[Pi]&quot;]]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;z&quot;]], &quot;\[Pi]&quot;], &quot;+&quot;, FractionBox[&quot;1&quot;, &quot;2&quot;]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[TagBox[RowBox[List[TagBox[FractionBox[&quot;1&quot;, &quot;2&quot;], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[&quot;3&quot;, &quot;2&quot;], &quot;-&quot;, FractionBox[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;z&quot;]], &quot;\[Pi]&quot;]]], HypergeometricPFQ, Rule[Editable, True]], &quot;,&quot;, TagBox[RowBox[List[FractionBox[RowBox[List[&quot;\[ImaginaryI]&quot;, &quot; &quot;, &quot;z&quot;]], &quot;\[Pi]&quot;], &quot;+&quot;, FractionBox[&quot;3&quot;, &quot;2&quot;]]], HypergeometricPFQ, Rule[Editable, True]]]], InterpretTemplate[Function[List[SlotSequence[1]]]]], HypergeometricPFQ, Rule[Editable, False]], &quot;;&quot;, TagBox[RowBox[List[&quot;-&quot;, &quot;1&quot;]], HypergeometricPFQ, Rule[Editable, True]]]], &quot;)&quot;]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], HypergeometricPFQ] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sech /> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 4 </cn> <pi /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> HypergeometricPFQ </ci> <list> <cn type='integer'> 1 </cn> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <plus /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </list> <list> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <cn type='rational'> 3 <sep /> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 3 <sep /> 2 </cn> </apply> </list> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sech", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["4", " ", "\[Pi]"]], ")"]], " ", RowBox[List["HypergeometricPFQ", "[", RowBox[List[RowBox[List["{", RowBox[List["1", ",", FractionBox["3", "2"], ",", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Pi]"]]], ",", RowBox[List[FractionBox["1", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Pi]"]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox["1", "2"], ",", RowBox[List[FractionBox["3", "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Pi]"]]], ",", RowBox[List[FractionBox["3", "2"], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "z"]], "\[Pi]"]]]]], "}"]], ",", RowBox[List["-", "1"]]]], "]"]]]], RowBox[List[SuperscriptBox["\[Pi]", "2"], "+", RowBox[List["4", " ", SuperscriptBox["z", "2"]]]]]]]]]]










Contributed by





Brychkov Yu.A. (2005)


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





2007-05-02