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ArcCoth




Mathematica Notation

Traditional Notation









Elementary Functions > ArcCoth[z] > Representations through equivalent functions > With related functions > Involving sech-1 > Involving coth-1((z2-1)1/2/(z2)1/2) > Involving coth-1((z2-1)1/2/(z2)1/2) and sech-1(1/z)





http://functions.wolfram.com/01.28.27.3494.01









  


  








Input Form





ArcCoth[Sqrt[z^2 - 1]/Sqrt[z^2]] == Sqrt[z/(z + 1)] Sqrt[(z + 1)/z] ArcSech[1/z] + (Sqrt[1 - z^2] Sqrt[1/(1 - z^2)] - (I Sqrt[z^2 - z^4])/(z Sqrt[-1 + z^2]) - 1) ((Pi I)/2)








Standard Form





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MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> coth </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <msqrt> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </mfrac> </msqrt> </mrow> <mo> - </mo> <mfrac> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </msqrt> </mrow> <mrow> <mi> z </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> + </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> </mfrac> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mi> &#8520; </mi> </mrow> <mn> 2 </mn> </mfrac> </mrow> <mo> + </mo> <mrow> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccoth /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <apply> <times /> <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> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <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='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <times /> <pi /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arcsech /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCoth", "[", FractionBox[SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "-", "1"]]], SqrtBox[SuperscriptBox["z_", "2"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SqrtBox[FractionBox["z", RowBox[List["z", "+", "1"]]]], " ", SqrtBox[FractionBox[RowBox[List["z", "+", "1"]], "z"]], " ", RowBox[List["ArcSech", "[", FractionBox["1", "z"], "]"]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]], "-", FractionBox[RowBox[List["\[ImaginaryI]", " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", SuperscriptBox["z", "4"]]]]]], RowBox[List["z", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]]]]], "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List["\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]]]]]]]]


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





2003-09-04