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Sech




Mathematica Notation

Traditional Notation









Elementary Functions > Sech[z] > Transformations > Products, sums, and powers of the direct function > Sums involving the direct function > Involving trigonometric functions > Involving csc





http://functions.wolfram.com/01.24.16.0050.01









  


  








Input Form





Sech[a] - Csc[b] == -2 Cosh[(a + I b)/2 + (I Pi)/4] Cosh[(a - I b)/2 - (I Pi)/4] Csc[b] Sech[a]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Sech", "[", "a", "]"]], "-", RowBox[List["Csc", "[", "b", "]"]]]], "\[Equal]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["Cosh", "[", RowBox[List[FractionBox[RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], "2"], " ", "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "4"]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List[FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], "2"], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "4"]]], "]"]], " ", RowBox[List["Csc", "[", "b", "]"]], " ", RowBox[List["Sech", "[", "a", "]"]]]]]]]]








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> sech </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> cosh </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> b </mi> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> csc </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sech </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <sech /> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <csc /> <ci> b </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <cosh /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <imaginaryi /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <cosh /> <apply> <plus /> <apply> <times /> <apply> <plus /> <ci> a </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> b </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <pi /> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <csc /> <ci> b </ci> </apply> <apply> <sech /> <ci> a </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["Sech", "[", "a_", "]"]], "-", RowBox[List["Csc", "[", "b_", "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]]]], "+", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "4"]]], "]"]], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], ")"]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], "4"]]], "]"]], " ", RowBox[List["Csc", "[", "b", "]"]], " ", RowBox[List["Sech", "[", "a", "]"]]]]]]]]


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





2003-08-21