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ArcCsch




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.29.27.0270.01









  


  








Input Form





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








Standard Form





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MathML Form







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</apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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


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





2003-08-21