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ArcCsc




Mathematica Notation

Traditional Notation









Elementary Functions > ArcCsc[z] > Representations through equivalent functions > With related functions > Involving tanh-1 > Involving csc-1(z) > Involving csc-1(z) and tanh-1(z2-2/2 (1-z2)1/2)





http://functions.wolfram.com/01.17.27.1481.01









  


  








Input Form





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








Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcCsc", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["\[Pi]", "4"], RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]]]], "-", RowBox[List[SqrtBox[RowBox[List["-", FractionBox["\[ImaginaryI]", "z"]]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]]]], "+", RowBox[List[SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]], " ", "z"]], "-", RowBox[List[SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]], "z"]], " ", SqrtBox[FractionBox["z", RowBox[List[RowBox[List["-", "1"]], "+", "z"]]]]]], "+", RowBox[List[SqrtBox[RowBox[List["1", "+", FractionBox["1", "z"]]]], " ", SqrtBox[FractionBox["z", RowBox[List["1", "+", "z"]]]]]]]], ")"]]]], "-", RowBox[List[FractionBox[SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", SuperscriptBox["z", "4"]]]], RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]]]]], " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[SuperscriptBox["z", "2"], "-", "2"]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]], "]"]]]]]]]]]]








MathML Form







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</mi> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </msqrt> </mrow> <mo> + </mo> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mi> z </mi> </mfrac> </mrow> </msqrt> <mo> &#8290; </mo> <msqrt> <mfrac> <mi> z </mi> <mrow> <mi> z </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </msqrt> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mfrac> <mrow> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> </msqrt> <mtext> </mtext> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <msqrt> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> tanh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mn> 2 </mn> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; 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Rule Form





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


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





2003-08-21