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ArcCosh






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.26.27.2312.01









  


  










Input Form





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










Standard Form





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










MathML Form







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










Rule Form





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










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





2003-08-21