Hyperbolic cosine integral: Transformations (formula 01.20.16.0026)

Cosh

$Mathematica Notation$

Elementary Functions

Transformations

Transformations and argument simplifications

Argument involving inverse trigonometric and hyperbolic functions

Involving coth^-1

http://functions.wolfram.com/01.20.16.0026.01

Input Form

Cosh[ArcCoth[z]/2] == (1/Sqrt[2]) Sqrt[1 + 1/Sqrt[1 - 1/z^2]]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["Cosh", "[", FractionBox[RowBox[List["ArcCoth", "[", "z", "]"]], "2"], "]"]], "\[Equal]", RowBox[List[FractionBox["1", SqrtBox["2"]], SqrtBox[RowBox[List["1", "+", FractionBox["1", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["z", "2"]]]]]]]]]]]]]]]

MathML Form

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

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Cosh", "[", FractionBox[RowBox[List["ArcCoth", "[", "z_", "]"]], "2"], "]"]], "]"]], "\[RuleDelayed]", FractionBox[SqrtBox[RowBox[List["1", "+", FractionBox["1", SqrtBox[RowBox[List["1", "-", FractionBox["1", SuperscriptBox["z", "2"]]]]]]]]], SqrtBox["2"]]]]]]

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

2001-10-29