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ArcSech






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.30.27.0358.01









  


  










Input Form





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










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ArcSech", "[", "z", "]"]], "\[Equal]", " ", RowBox[List[RowBox[List[FractionBox["\[Pi]", RowBox[List["4", " ", SqrtBox[FractionBox[RowBox[List[RowBox[List["-", "1"]], "+", "z"]], "z"]]]]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", FractionBox["1", "z"]]]], RowBox[List["(", RowBox[List["2", "-", FractionBox[RowBox[List[SqrtBox[RowBox[List[FractionBox["1", SuperscriptBox["z", "4"]], "-", FractionBox["1", SuperscriptBox["z", "2"]]]]], " ", "z"]], SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", FractionBox["1", SuperscriptBox["z", "2"]]]]]], "-", RowBox[List[SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]], " ", "z"]], "-", RowBox[List[SqrtBox[RowBox[List["1", "-", FractionBox[SqrtBox["2"], "z"]]]], " ", SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]], " ", SqrtBox[RowBox[List["-", "z"]]], SqrtBox[FractionBox["z", RowBox[List[RowBox[List["-", SqrtBox["2"]]], "+", "z"]]]]]], "+", RowBox[List[SqrtBox[FractionBox["1", "z"]], " ", SqrtBox["z"], " ", SqrtBox[FractionBox["z", RowBox[List[SqrtBox["2"], "+", "z"]]]], " ", SqrtBox[FractionBox[RowBox[List[SqrtBox["2"], "+", "z"]], "z"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox["2"], "-", "z"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "z"]], ")"]]]]], " ", SqrtBox[RowBox[List[SqrtBox["2"], "+", "z"]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[FractionBox["1", SuperscriptBox["z", "4"]], "-", FractionBox["1", SuperscriptBox["z", "2"]]]]], " ", SuperscriptBox["z", RowBox[List["3", "/", "2"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "2"]], "+", SuperscriptBox["z", "2"]]]]]]]]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List["1", "+", "z"]], "z"]]]], RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", RowBox[List["ArcCos", "[", FractionBox[RowBox[List["2", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], "-", "1"]]]]], SuperscriptBox["z", "2"]], "]"]]]], ")"]]]]]]]]]]










MathML Form







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</apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <ci> z </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> <apply> <power /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> 1 </cn> <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> <plus /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </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 /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <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 /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <plus /> <ci> z </ci> <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> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <plus /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <plus /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -2 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <plus /> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <arccos /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <plus /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </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> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





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










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





2003-08-21