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ArcCosh




Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/01.26.27.1379.01









  


  








Input Form





ArcCosh[2 z Sqrt[1 - z^2]] == (Sqrt[2 z Sqrt[1 - z^2] - 1]/ Sqrt[1 - 2 z Sqrt[1 - z^2]]) ((-((2 Sqrt[1 - 2 z^2] Sqrt[-z^2 + z^4])/(Sqrt[-z^2] Sqrt[-1 + z^2] Sqrt[-1 + 2 z^2]))) ArcSec[1/z] + (Pi/2) (1 + ((Sqrt[1 - 2 z^2] Sqrt[z^2 (-1 + z^2)])/ ((1 - z^2) z^3 Sqrt[-z^2] Sqrt[-1 + 2 z^2])) ((z^2)^(3/2) Sqrt[-1 + z^2] - Sqrt[1/z] z^(7/2) Sqrt[1/(1 + Sqrt[2] z)] Sqrt[1 + Sqrt[2] z] Sqrt[-1 + z^2] + z^2 Sqrt[z^2 (-1 + z^2)] + z^3 Sqrt[-1 + z^2] (-2 + Sqrt[-(1/z)] Sqrt[-z] Sqrt[1/(1 - Sqrt[2] z)] Sqrt[1 - Sqrt[2] z]))))








Standard Form





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MathML Form







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type='integer'> -1 </cn> </apply> </apply> <apply> <arcsec /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCosh", "[", RowBox[List["2", " ", "z_", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z_", "2"]]]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]], "-", "1"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", SuperscriptBox["z", "2"]]], "+", SuperscriptBox["z", "4"]]]]]], ")"]], " ", RowBox[List["ArcSec", "[", FractionBox["1", "z"], "]"]]]], RowBox[List[SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]], " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["(", SuperscriptBox["z", "2"], ")"]], RowBox[List["3", "/", "2"]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]]]], "-", RowBox[List[SqrtBox[FractionBox["1", "z"]], " ", SuperscriptBox["z", RowBox[List["7", "/", "2"]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "+", RowBox[List[SqrtBox["2"], " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "+", RowBox[List[SqrtBox["2"], " ", "z"]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]]]], "+", RowBox[List[SuperscriptBox["z", "2"], " ", SqrtBox[RowBox[List[SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]]]]]]], "+", RowBox[List[SuperscriptBox["z", "3"], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List[SqrtBox[RowBox[List["-", FractionBox["1", "z"]]]], " ", SqrtBox[RowBox[List["-", "z"]]], " ", SqrtBox[FractionBox["1", RowBox[List["1", "-", RowBox[List[SqrtBox["2"], " ", "z"]]]]]], " ", SqrtBox[RowBox[List["1", "-", RowBox[List[SqrtBox["2"], " ", "z"]]]]]]]]], ")"]]]]]], ")"]]]], RowBox[List[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], " ", SuperscriptBox["z", "3"], " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "1"]], "+", RowBox[List["2", " ", SuperscriptBox["z", "2"]]]]]]]]]]], ")"]]]]]], ")"]]]], SqrtBox[RowBox[List["1", "-", RowBox[List["2", " ", "z", " ", SqrtBox[RowBox[List["1", "-", SuperscriptBox["z", "2"]]]]]]]]]]]]]]


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





2003-08-21