Inverse hyperbolic cosine: Representations through equivalent functions (formula 01.26.27.2419)

ArcCosh

$Mathematica Notation$

Elementary Functions

Representations through equivalent functions

With related functions

Involving sech^-1

Involving cosh^-1((a-z)^1/2/(-2 z)^1/2)

Involving cosh^-1((-z-1)^1/2/(-2 z)^1/2) and sech^-1(z)

http://functions.wolfram.com/01.26.27.2419.01

Input Form

ArcCosh[Sqrt[-z - 1]/Sqrt[-2 z]] == (1/2) ArcSech[z]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["ArcCosh", "[", FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "z"]], "-", "1"]]], SqrtBox[RowBox[List[RowBox[List["-", "2"]], " ", "z"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox["1", "2"], RowBox[List["ArcSech", "[", "z", "]"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msup> <mi> cosh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msqrt> <msqrt> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msqrt> </mfrac> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> sech </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <arccosh /> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> z </ci> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <arcsech /> <ci> z </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ArcCosh", "[", FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "z_"]], "-", "1"]]], SqrtBox[RowBox[List[RowBox[List["-", "2"]], " ", "z_"]]]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List["ArcSech", "[", "z", "]"]], "2"]]]]]

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

2003-08-21