Hyperbolic cosecant: Transformations (formula 01.23.16.0051)

Csch

$Mathematica Notation$

Elementary Functions

Transformations

Products, sums, and powers of the direct function

Sums involving the direct function

Involving trigonometric functions

http://functions.wolfram.com/01.23.16.0051.01

Input Form

Csch[a] - I Csc[b] == 2 Cosh[(a - I b)/2] Sin[(b - I a)/2] Csc[b] Csch[a]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Csch", "[", "a", "]"]], "-", RowBox[List["\[ImaginaryI]", " ", RowBox[List["Csc", "[", "b", "]"]]]]]], "\[Equal]", RowBox[List["2", " ", RowBox[List["Cosh", "[", FractionBox[RowBox[List["a", "-", RowBox[List["\[ImaginaryI]", " ", "b"]]]], "2"], "]"]], RowBox[List["Sin", "[", FractionBox[RowBox[List["b", "-", RowBox[List["\[ImaginaryI]", " ", "a"]]]], "2"], "]"]], " ", RowBox[List["Csc", "[", "b", "]"]], " ", RowBox[List["Csch", "[", "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> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> a </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> b </mi> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> b </mi> <mo> - </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mi> csc </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mi> b </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> a </mi> <mo> ) </mo> </mrow> <mtext> </mtext> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <csch /> <ci> a </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <csc /> <ci> b </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <cosh /> <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> <apply> <sin /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <ci> csc </ci> <ci> b </ci> <apply> <csch /> <ci> a </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

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

2003-08-21