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Coth




Mathematica Notation

Traditional Notation









Elementary Functions > Coth[z] > Transformations > Transformations and argument simplifications > Argument involving inverse trigonometric and hyperbolic functions > Involving cos-1





http://functions.wolfram.com/01.22.16.0067.01









  


  








Input Form





Coth[a ArcCos[z]] == 1 + 2/(-1 + E^(a Pi) (I z + Sqrt[1 - z^2])^(2 a I))








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["Coth", "[", RowBox[List["a", " ", RowBox[List["ArcCos", "[", "z", "]"]]]], "]"]], "\[Equal]", RowBox[List["1", "+", FractionBox["2", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", "\[Pi]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["2", " ", "a", " ", "\[ImaginaryI]"]]]]]]]]]]]]]]








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> coth </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> cos </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 2 </mn> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mi> a </mi> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> a </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> </msup> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <coth /> <apply> <times /> <ci> a </ci> <apply> <arccos /> <ci> z </ci> </apply> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> a </ci> <pi /> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <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> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <imaginaryi /> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </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["Coth", "[", RowBox[List["a_", " ", RowBox[List["ArcCos", "[", "z_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List["1", "+", FractionBox["2", RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", "\[Pi]"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["\[ImaginaryI]", " ", "z"]], "+", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], ")"]], RowBox[List["2", " ", "a", " ", "\[ImaginaryI]"]]]]]]]]]]]]]]


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





2007-05-02