Logarithmic integral: Representations through equivalent functions (formula 06.36.27.0004)

LogIntegral

$Mathematica Notation$

Gamma, Beta, Erf

LogIntegral[z]

Representations through equivalent functions

With related functions

http://functions.wolfram.com/06.36.27.0004.01

Input Form

LogIntegral[z] == CoshIntegral[Log[z]] + SinhIntegral[Log[z]] - (1/2) (Log[1/Log[z]] + Log[Log[z]])

Standard Form

Cell[BoxData[RowBox[List[RowBox[List["LogIntegral", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["CoshIntegral", "[", RowBox[List["Log", "[", "z", "]"]], "]"]], "+", RowBox[List["SinhIntegral", "[", RowBox[List["Log", "[", "z", "]"]], "]"]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", FractionBox["1", RowBox[List["Log", "[", "z", "]"]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List["Log", "[", "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> <mi> li </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> Chi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> Shi </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> LogIntegral </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> CoshIntegral </ci> <apply> <ln /> <ci> z </ci> </apply> </apply> <apply> <ci> SinhIntegral </ci> <apply> <ln /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <ln /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ln /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ln /> <apply> <ln /> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LogIntegral", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["CoshIntegral", "[", RowBox[List["Log", "[", "z", "]"]], "]"]], "+", RowBox[List["SinhIntegral", "[", RowBox[List["Log", "[", "z", "]"]], "]"]], "-", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", FractionBox["1", RowBox[List["Log", "[", "z", "]"]]], "]"]], "+", RowBox[List["Log", "[", RowBox[List["Log", "[", "z", "]"]], "]"]]]], ")"]]]]]]]]]]

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

2007-05-02