Cosecant: Transformations (formula 01.10.16.0081)

Csc

$Mathematica Notation$

Elementary Functions

Transformations

Transformations and argument simplifications

Argument involving inverse trigonometric and hyperbolic functions

Involving tanh^-1

http://functions.wolfram.com/01.10.16.0081.01

Input Form

Csc[a ArcTanh[z]] == -((2 I (1 - z^2)^((I a)/2))/ ((1 - z)^(I a) - (1 + z)^(I a)))

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Csc", "[", RowBox[List["a", " ", RowBox[List["ArcTanh", "[", "z", "]"]]]], "]"]], "\[Equal]", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "2"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[ImaginaryI]", " ", "a"]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["\[ImaginaryI]", " ", "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> <mi> csc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo>  </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> ⅈ </mi> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> <mn> 2 </mn> </mfrac> </msup> </mrow> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msup> <mo> - </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> a </mi> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <csc /> <apply> <times /> <ci> a </ci> <apply> <arctanh /> <ci> z </ci> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <imaginaryi /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <times /> <imaginaryi /> <ci> a </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <imaginaryi /> <ci> a </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Csc", "[", RowBox[List["a_", " ", RowBox[List["ArcTanh", "[", "z_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["-", FractionBox[RowBox[List["2", " ", "\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], FractionBox[RowBox[List["\[ImaginaryI]", " ", "a"]], "2"]]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", "z"]], ")"]], RowBox[List["\[ImaginaryI]", " ", "a"]]], "-", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", "z"]], ")"]], RowBox[List["\[ImaginaryI]", " ", "a"]]]]]]]]]]]]

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

2007-05-02