Hyperbolic cosine integral: Transformations (formula 01.20.16.0068)

Cosh

$Mathematica Notation$

Elementary Functions

Cosh[z]

Transformations

Transformations and argument simplifications

Argument involving inverse trigonometric and hyperbolic functions

Involving sinh^-1

http://functions.wolfram.com/01.20.16.0068.01

Input Form

Cosh[n (ArcSinh[z]/2)] == ChebyshevT[n, Sqrt[1 + Sqrt[1 + z^2]]/Sqrt[2]] /; Element[n, Integers] && n > 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Cosh", "[", RowBox[List["n", " ", FractionBox[RowBox[List["ArcSinh", "[", "z", "]"]], "2"]]], "]"]], "\[Equal]", RowBox[List["ChebyshevT", "[", RowBox[List["n", ",", FractionBox[SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]]], SqrtBox["2"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["Element", "[", RowBox[List["n", ",", "Integers"]], "]"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> cosh </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <msup> <mi> sinh </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <msub> <mi> T </mi> <mi> n </mi> </msub> <mo> ( </mo> <mfrac> <msqrt> <mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> </msqrt> <mo> + </mo> <mn> 1 </mn> </mrow> </msqrt> <msqrt> <mn> 2 </mn> </msqrt> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <cosh /> <apply> <times /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <arcsinh /> <ci> z </ci> </apply> </apply> </apply> <apply> <ci> ChebyshevT </ci> <ci> n </ci> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> 1 </cn> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <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> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Cosh", "[", RowBox[List[FractionBox["1", "2"], " ", "n_", " ", RowBox[List["ArcSinh", "[", "z_", "]"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["ChebyshevT", "[", RowBox[List["n", ",", FractionBox[SqrtBox[RowBox[List["1", "+", SqrtBox[RowBox[List["1", "+", SuperscriptBox["z", "2"]]]]]]], SqrtBox["2"]]]], "]"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

2001-10-29