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ArcCsch






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.29.27.2109.01









  


  










Input Form





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










Standard Form





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










MathML Form







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Rule Form





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










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





2003-08-21