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http://functions.wolfram.com/06.36.06.0025.01
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LogIntegral[z] \[Proportional]
(z/Log[z]) (1 + 1/Log[z] + 2/Log[z]^2 + O[1/Log[z]^3]) +
I Pi (1 + 2 Floor[-(Arg[-z]/(2 Pi))] + Floor[(Pi + Arg[-z])/(2 Pi)]) /;
(Abs[z] -> Infinity || z -> 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LogIntegral", "[", "z", "]"]], "\[Proportional]", RowBox[List[RowBox[List[FractionBox["z", RowBox[List["Log", "[", "z", "]"]]], RowBox[List["(", RowBox[List["1", "+", FractionBox["1", RowBox[List["Log", "[", "z", "]"]]], "+", FractionBox["2", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "2"]], "+", RowBox[List["O", "[", FractionBox["1", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "3"]], "]"]]]], ")"]]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List["Arg", "[", RowBox[List["-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["-", "z"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", RowBox[List[RowBox[List["\[Infinity]", "\[Or]", "z"]], "\[Rule]", "0"]]]], ")"]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mi> li </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mrow> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mo> + </mo> <mi> π </mi> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> <mo> ⌋ </mo> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <mi> arg </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mo> - </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> </mrow> </mfrac> </mrow> <mo> ⌋ </mo> </mrow> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mfrac> <mi> z </mi> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mfrac> <mn> 1 </mn> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> <mo> + </mo> <mfrac> <mn> 2 </mn> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> <mo> + </mo> <mrow> <mi> O </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mn> 1 </mn> <mrow> <msup> <mi> log </mi> <mn> 3 </mn> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[LeftBracketingBar]" </annotation> </semantics> <mi> z </mi> <semantics> <mo> ❘ </mo> <annotation encoding='Mathematica'> "\[RightBracketingBar]" </annotation> </semantics> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mrow> <mrow> <mi> ∞ </mi> <mo> ∨ </mo> <mi> z </mi> </mrow> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 0 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> LogIntegral </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <imaginaryi /> <pi /> <apply> <plus /> <apply> <floor /> <apply> <times /> <apply> <plus /> <apply> <arg /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <pi /> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <arg /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <ln /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ln /> <ci> z </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <power /> <apply> <ln /> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <ln /> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <ci> Rule </ci> <apply> <abs /> <ci> z </ci> </apply> <apply> <ci> Rule </ci> <apply> <or /> <infinity /> <ci> z </ci> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LogIntegral", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List["1", "+", FractionBox["1", RowBox[List["Log", "[", "z", "]"]]], "+", FractionBox["2", SuperscriptBox[RowBox[List["Log", "[", "z", "]"]], "2"]], "+", RowBox[List["SeriesData", "[", RowBox[List[RowBox[List["Log", "[", "z", "]"]], ",", "\[Infinity]", ",", RowBox[List["{", "0", "}"]], ",", "0", ",", "3"]], "]"]]]], ")"]]]], RowBox[List["Log", "[", "z", "]"]]], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List["2", " ", RowBox[List["Floor", "[", RowBox[List["-", FractionBox[RowBox[List["Arg", "[", RowBox[List["-", "z"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]], "+", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "+", RowBox[List["Arg", "[", RowBox[List["-", "z"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", RowBox[List[RowBox[List["\[Infinity]", "||", "z"]], "\[Rule]", "0"]]]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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