Hyperbolic cosecant: Transformations (formula 01.23.16.0047)

Csch

$Mathematica Notation$

Elementary Functions

Transformations

Products, sums, and powers of the direct function

Sums involving the direct function

Involving other hyperbolic functions

http://functions.wolfram.com/01.23.16.0047.01

Input Form

a Csch[z] + b Sech[z] == 2 Sqrt[1 - a^2/b^2] b Csch[2 z] Sinh[z + ArcTanh[a/b]]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["a", " ", RowBox[List["Csch", "[", "z", "]"]]]], "+", RowBox[List["b", " ", RowBox[List["Sech", "[", "z", "]"]]]]]], "\[Equal]", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["a", "2"], SuperscriptBox["b", "2"]]]]], " ", "b", " ", RowBox[List["Csch", "[", RowBox[List["2", " ", "z"]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["z", "+", RowBox[List["ArcTanh", "[", FractionBox["a", "b"], "]"]]]], "]"]]]]]]]]

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> a </mi> <mo> ⁢ </mo> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> sech </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mfrac> <msup> <mi> a </mi> <mn> 2 </mn> </msup> <msup> <mi> b </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msqrt> <mo> ⁢ </mo> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mi> csch </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> sinh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mi> a </mi> <mi> b </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <csch /> <ci> z </ci> </apply> </apply> <apply> <times /> <ci> b </ci> <apply> <sech /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> b </ci> <apply> <csch /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <sinh /> <apply> <plus /> <ci> z </ci> <apply> <arctanh /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> b </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[RowBox[List["a_", " ", RowBox[List["Csch", "[", "z_", "]"]]]], "+", RowBox[List["b_", " ", RowBox[List["Sech", "[", "z_", "]"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", FractionBox[SuperscriptBox["a", "2"], SuperscriptBox["b", "2"]]]]], " ", "b", " ", RowBox[List["Csch", "[", RowBox[List["2", " ", "z"]], "]"]], " ", RowBox[List["Sinh", "[", RowBox[List["z", "+", RowBox[List["ArcTanh", "[", FractionBox["a", "b"], "]"]]]], "]"]]]]]]]]

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

2003-08-21