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ArcCosh






Mathematica Notation

Traditional Notation









Elementary Functions > ArcCosh[z] > Integral representations > Contour integral representations





http://functions.wolfram.com/01.26.07.0004.01









  


  










Input Form





ArcCosh[z] == (Sqrt[z - 1]/Sqrt[1 - z]) (Pi/2 - (z/(2 Sqrt[Pi])) (1/(2 Pi I)) ContourIntegrate[ (Gamma[s] Gamma[1/2 + s] Gamma[1/2 - s]^2)/(1 - z^2)^s, {s, \[ScriptCapitalL]}]) /; Abs[Arg[1 - z^2]] < Pi










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCosh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["z", "-", "1"]]], " ", RowBox[List["(", RowBox[List[FractionBox["\[Pi]", "2"], "-", FractionBox[RowBox[List["z", " ", RowBox[List["ContourIntegrate", "[", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "s", "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "s"]], "]"]], " ", SuperscriptBox[RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "-", "s"]], "]"]], "2"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], ")"]], RowBox[List["-", "s"]]]]], ",", RowBox[List["{", RowBox[List["s", ",", "\[ScriptCapitalL]"]], "}"]]]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List["2", " ", SqrtBox["\[Pi]"]]], ")"]], " ", RowBox[List["(", RowBox[List["2", " ", "\[Pi]", " ", "\[ImaginaryI]"]], ")"]]]]]]], ")"]]]], SqrtBox[RowBox[List["1", "-", "z"]]]], "/;", RowBox[List[RowBox[List["Abs", "[", RowBox[List["Arg", "[", RowBox[List["1", "-", SuperscriptBox["z", "2"]]], "]"]], "]"]], "<", "\[Pi]"]]]]]]]]










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





2001-10-29