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ArcSech




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.30.27.2651.01









  


  








Input Form





ArcSech[z^2/(2 Sqrt[-1 - z^2])] == (Sqrt[(2 Sqrt[-1 - z^2])/z^2 - 1]/Sqrt[1 - (2 Sqrt[-1 - z^2])/z^2]) (Pi/2 - ((z^3 Sqrt[-2 - z^2] Sqrt[-1 - z^2])/(2 Sqrt[1 - I z] (1 + I z) Sqrt[-2 - 3 z^2 - z^4])) Sqrt[(1 + z^2)/z^4] Sqrt[(I - z)/z] Sqrt[-(I/z)] (Pi (Sqrt[-(1/z^2)] z - I Sqrt[(z + I Sqrt[2])/z] Sqrt[I/z] Sqrt[(-I) z] Sqrt[z/(I Sqrt[2] + z)] + I Sqrt[(z - I Sqrt[2])/z] Sqrt[-(I/z)] Sqrt[I z] Sqrt[z/((-I) Sqrt[2] + z)] - (Sqrt[-((1 + z^2)/z^2)] Sqrt[(1 + z^2)/z^4] z^3)/(1 + z^2)) + 4 ArcCsch[z]))








Standard Form





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MathML Form







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<times /> <cn type='integer'> -1 </cn> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <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 /> <imaginaryi /> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> z </ci> </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 /> <imaginaryi /> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> z </ci> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <imaginaryi /> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <imaginaryi /> </apply> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <apply> <times /> <imaginaryi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <arccsch /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcSech", "[", FractionBox[SuperscriptBox["z_", "2"], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z_", "2"]]]]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox[RowBox[List[FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]], SuperscriptBox["z", "2"]], "-", "1"]]], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["z", "3"], " ", SqrtBox[RowBox[List[RowBox[List["-", "2"]], "-", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]], ")"]], " ", SqrtBox[FractionBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "4"]]], " ", SqrtBox[FractionBox[RowBox[List["\[ImaginaryI]", "-", "z"]], "z"]], " ", SqrtBox[RowBox[List["-", FractionBox["\[ImaginaryI]", "z"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["\[Pi]", " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[RowBox[List["-", FractionBox["1", SuperscriptBox["z", "2"]]]]], " ", "z"]], "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox[RowBox[List["z", "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"]]]]], "z"]], " ", SqrtBox[FractionBox["\[ImaginaryI]", "z"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", "z"]]], " ", SqrtBox[FractionBox["z", RowBox[List[RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"]]], "+", "z"]]]]]], "+", RowBox[List["\[ImaginaryI]", " ", SqrtBox[FractionBox[RowBox[List["z", "-", RowBox[List["\[ImaginaryI]", " ", SqrtBox["2"]]]]], "z"]], " ", SqrtBox[RowBox[List["-", FractionBox["\[ImaginaryI]", "z"]]]], " ", SqrtBox[RowBox[List["\[ImaginaryI]", " ", "z"]]], " ", SqrtBox[FractionBox["z", RowBox[List[RowBox[List[RowBox[List["-", "\[ImaginaryI]"]], " ", SqrtBox["2"]]], "+", "z"]]]]]], "-", FractionBox[RowBox[List[SqrtBox[RowBox[List["-", FractionBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "2"]]]]], " ", SqrtBox[FractionBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]], SuperscriptBox["z", "4"]]], " ", SuperscriptBox["z", "3"]]], RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]], ")"]]]], "+", RowBox[List["4", " ", RowBox[List["ArcCsch", "[", "z", "]"]]]]]], ")"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", RowBox[List["\[ImaginaryI]", " ", "z"]]]]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["\[ImaginaryI]", " ", "z"]]]], ")"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "2"]], "-", RowBox[List["3", " ", SuperscriptBox["z", "2"]]], "-", SuperscriptBox["z", "4"]]]]]]]]], ")"]]]], SqrtBox[RowBox[List["1", "-", FractionBox[RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "-", SuperscriptBox["z", "2"]]]]]], SuperscriptBox["z", "2"]]]]]]]]]]


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





2003-08-21