Inverse hyperbolic cosine: Representations through equivalent functions (formula 01.26.27.0027)

ArcCosh

$Mathematica Notation$

Elementary Functions

ArcCosh[z]

Representations through equivalent functions

With related functions

Involving cot^-1

Involving cosh^-1(z/(z²+1)^1/2)

Involving cosh^-1(z/(z²+1)^1/2) and cot^-1(z)

http://functions.wolfram.com/01.26.27.0027.01

Input Form

ArcCosh[z/Sqrt[z^2 + 1]] == (Sqrt[z - 1]/Sqrt[1 - z]) ArcCot[z] + ((Pi Sqrt[-z^2])/(2 z)) (Sqrt[1/z^2] z - 1) /; Abs[z] < 1 || ( !IntervalMemberQ[Interval[{-Infinity, -1}], z] && !IntervalMemberQ[Interval[{1, Infinity}], z] && !IntervalMemberQ[Interval[{-Infinity, -1}], I z] && !IntervalMemberQ[Interval[{1, Infinity}], I z])

Standard Form

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MathML Form

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Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCosh", "[", FractionBox["z_", SqrtBox[RowBox[List[SuperscriptBox["z_", "2"], "+", "1"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SqrtBox[RowBox[List["z", "-", "1"]]], " ", RowBox[List["ArcCot", "[", "z", "]"]]]], SqrtBox[RowBox[List["1", "-", "z"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["\[Pi]", " ", SqrtBox[RowBox[List["-", SuperscriptBox["z", "2"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[SqrtBox[FractionBox["1", SuperscriptBox["z", "2"]]], " ", "z"]], "-", "1"]], ")"]]]], RowBox[List["2", " ", "z"]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", "1"]], "||", RowBox[List["(", RowBox[List[RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", "z"]], "]"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", "z"]], "]"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", "\[Infinity]"]], ",", RowBox[List["-", "1"]]]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]], "&&", RowBox[List["!", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List["1", ",", "\[Infinity]"]], "}"]], "]"]], ",", RowBox[List["\[ImaginaryI]", " ", "z"]]]], "]"]]]]]], ")"]]]]]]]]]]

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

2001-10-29