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ArcCsc




Mathematica Notation

Traditional Notation









Elementary Functions > ArcCsc[z] > Representations through equivalent functions > With related functions > Involving coth-1 > Involving csc-1(z2-1/z2+1) > Involving csc-1(z2-1/z2+1) and coth-1(z)





http://functions.wolfram.com/01.17.27.1826.01









  


  








Input Form





ArcCsc[(z^2 - 1)/(z^2 + 1)] == (-(1/2)) Pi (3 - 2 Sqrt[1/(1 - z)] Sqrt[1 - z] Sqrt[-(1/z)] Sqrt[-z] - 2 Sqrt[1/z] Sqrt[z] Sqrt[1/(1 + z)] Sqrt[1 + z]) + ((2 Sqrt[-z^2])/z) Sqrt[(1 - z)/(1 + z)] Sqrt[(1 + z)/(1 - z)] ArcCoth[z]








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcCsc", "[", FractionBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]], RowBox[List[SuperscriptBox["z", "2"], "+", "1"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", "z"]]]], " ", SqrtBox[RowBox[List["1", "-", "z"]]], " ", SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]], " ", SqrtBox[RowBox[List["-", "z"]]]]], "-", RowBox[List["2", " ", SqrtBox[FractionBox["1", "z"]], " ", SqrtBox["z"], " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", "z"]]]], " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " "]], "z"], SqrtBox[FractionBox[RowBox[List["1", "-", "z"]], RowBox[List["1", "+", "z"]]]], " ", SqrtBox[FractionBox[RowBox[List["1", "+", "z"]], RowBox[List["1", "-", "z"]]]], RowBox[List["ArcCoth", "[", "z", "]"]]]]]]]]]]








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> csc </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8290; </mo> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 3 </mn> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <mi> z </mi> </mrow> </msqrt> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mi> z </mi> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mn> 1 </mn> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mo> - </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> </mrow> <mi> z </mi> </mfrac> <mo> &#8290; </mo> <msqrt> <mfrac> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> <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> <mrow> <mn> 1 </mn> <mo> - </mo> <mi> z </mi> </mrow> </mfrac> </msqrt> <mo> &#8290; </mo> <mrow> <msup> <mi> coth </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccsc /> <apply> <times /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <pi /> <apply> <plus /> <cn type='integer'> 3 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <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> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <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> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 2 </cn> <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> <power /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <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 /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <arccoth /> <ci> z </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCsc", "[", FractionBox[RowBox[List[SuperscriptBox["z_", "2"], "-", "1"]], RowBox[List[SuperscriptBox["z_", "2"], "+", "1"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["3", "-", RowBox[List["2", " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", "z"]]]], " ", SqrtBox[RowBox[List["1", "-", "z"]]], " ", SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]], " ", SqrtBox[RowBox[List["-", "z"]]]]], "-", RowBox[List["2", " ", SqrtBox[FractionBox["1", "z"]], " ", SqrtBox["z"], " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", "z"]]]], " ", SqrtBox[RowBox[List["1", "+", "z"]]]]]]], ")"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["1", "-", "z"]], RowBox[List["1", "+", "z"]]]], " ", SqrtBox[FractionBox[RowBox[List["1", "+", "z"]], RowBox[List["1", "-", "z"]]]], " ", RowBox[List["ArcCoth", "[", "z", "]"]]]], "z"]]]]]]]


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





2003-08-21