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Sec




Mathematica Notation

Traditional Notation









Elementary Functions > Sec[z] > Transformations > Products, sums, and powers of the direct function > Sums involving the direct function > Involving hyperbolic functions > Involving csch





http://functions.wolfram.com/01.11.16.0050.01









  


  








Input Form





Sec[z] + I Csch[z] == 2 I Cos[(z E^((I Pi)/4))/Sqrt[2] + Pi/4] Cos[z/(E^((I Pi)/4) Sqrt[2]) - Pi/4] Csch[z] Sec[z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Sec", "[", "z", "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Csch", "[", "z", "]"]]]]]], "\[Equal]", RowBox[List["2", "\[ImaginaryI]", " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["z", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "4"]]]], SqrtBox["2"]], "+", FractionBox["\[Pi]", "4"]]], "]"]], RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["z", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "4"]]]]]], SqrtBox["2"]], "-", FractionBox["\[Pi]", "4"]]], "]"]], RowBox[List["Csch", "[", "z", "]"]], " ", RowBox[List["Sec", "[", "z", "]"]]]]]]]]








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> <mi> sec </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> &#8520; </mi> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <msup> <mi> &#8519; </mi> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 4 </mn> </mfrac> </msup> <mo> &#8290; </mo> <mi> z </mi> </mrow> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> <mo> + </mo> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> z </mi> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> <mo> - </mo> <mfrac> <mi> &#960; </mi> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> csch </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sec </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <sec /> <ci> z </ci> </apply> <apply> <times /> <imaginaryi /> <apply> <csch /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <cos /> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <imaginaryi /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> z </ci> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <cos /> <apply> <plus /> <apply> <times /> <ci> z </ci> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <times /> <imaginaryi /> <pi /> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <csch /> <ci> z </ci> </apply> <apply> <sec /> <ci> z </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["Sec", "[", "z_", "]"]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Csch", "[", "z_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["2", " ", "\[ImaginaryI]", " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["z", " ", SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "4"]]]], SqrtBox["2"]], "+", FractionBox["\[Pi]", "4"]]], "]"]], " ", RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List["z", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]]]]]]], SqrtBox["2"]], "-", FractionBox["\[Pi]", "4"]]], "]"]], " ", RowBox[List["Csch", "[", "z", "]"]], " ", RowBox[List["Sec", "[", "z", "]"]]]]]]]]


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





2003-08-21