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ArcCsch






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.29.27.1504.01









  


  










Input Form





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










Standard Form





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MathML Form







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<times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <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 /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> 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<power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn 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</math>










Rule Form





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










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





2003-08-21