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ArcCoth




Mathematica Notation

Traditional Notation









Elementary Functions > ArcCoth[z] > Representations through equivalent functions > With related functions > Involving tan-1 > Involving coth-1(z-1/2(-z)1/2) > Involving coth-1(z-1/2(-z)1/2) and tan-1(z1/2)





http://functions.wolfram.com/01.28.27.0929.01









  


  








Input Form





ArcCoth[(z - 1)/(2 Sqrt[-z])] == (-((2 Sqrt[-z])/Sqrt[z])) ArcTan[Sqrt[z]] + (Pi Sqrt[-z^2])/z /; Abs[z] > 1








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ArcCoth", "[", FractionBox[RowBox[List["z", "-", "1"]], RowBox[List["2", SqrtBox[RowBox[List["-", "z"]]]]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List["2", SqrtBox[RowBox[List["-", "z"]]]]], SqrtBox["z"]]]], " ", RowBox[List["ArcTan", "[", SqrtBox["z"], "]"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], "z"]]]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]]]]]]








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> <msup> <mi> coth </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mi> z </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mtext> </mtext> </mrow> <msqrt> <mi> z </mi> </msqrt> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <msqrt> <mi> z </mi> </msqrt> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mfrac> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mi> z </mi> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[LeftBracketingBar]&quot; </annotation> </semantics> <mi> z </mi> <semantics> <mo> &#10072; </mo> <annotation encoding='Mathematica'> &quot;\[RightBracketingBar]&quot; </annotation> </semantics> </mrow> <mo> &gt; </mo> <mn> 1 </mn> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <arccoth /> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <arctan /> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <pi /> <apply> <power /> <apply> <times /> <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> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <gt /> <apply> <abs /> <ci> z </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCoth", "[", FractionBox[RowBox[List["z_", "-", "1"]], RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z_"]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox[RowBox[List["-", "z"]]]]], ")"]], " ", RowBox[List["ArcTan", "[", SqrtBox["z"], "]"]]]], SqrtBox["z"]]]], "+", FractionBox[RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], "z"]]], "/;", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], ">", "1"]]]]]]]]


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





2003-09-04