Hyperbolic cosine integral: Representations through equivalent functions (formula 01.20.27.0046)

Cosh

$Mathematica Notation$

Elementary Functions

Cosh[z]

Representations through equivalent functions

With related functions

Involving coth

http://functions.wolfram.com/01.20.27.0046.01

Input Form

Cosh[z] == (I Coth[z])/Sqrt[1 - Coth[z]^2] /; 0 < Im[z] < Pi

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Cosh", "[", "z", "]"]], "\[Equal]", FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Coth", "[", "z", "]"]], " "]], SqrtBox[RowBox[List["1", "-", SuperscriptBox[RowBox[List["Coth", "[", "z", "]"]], "2"]]]]]]], "/;", RowBox[List["0", "<", RowBox[List["Im", "[", "z", "]"]], "<", "\[Pi]"]]]]]]

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> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mfrac> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mrow> <mi> coth </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mn> 1 </mn> <mo> - </mo> <mrow> <msup> <mi> coth </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> </mfrac> </mrow> <mo> /; </mo> <mrow> <mn> 0 </mn> <mo> < </mo> <mrow> <mi> Im </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> < </mo> <mi> π </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <cosh /> <ci> z </ci> </apply> <apply> <times /> <imaginaryi /> <apply> <coth /> <ci> z </ci> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <apply> <coth /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <lt /> <cn type='integer'> 0 </cn> <apply> <imaginary /> <ci> z </ci> </apply> <pi /> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Cosh", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List["\[ImaginaryI]", " ", RowBox[List["Coth", "[", "z", "]"]]]], SqrtBox[RowBox[List["1", "-", SuperscriptBox[RowBox[List["Coth", "[", "z", "]"]], "2"]]]]], "/;", RowBox[List["0", "<", RowBox[List["Im", "[", "z", "]"]], "<", "\[Pi]"]]]]]]]]

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

2001-10-29