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LogIntegral






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > LogIntegral[z] > Complex characteristics > Real part





http://functions.wolfram.com/06.36.19.0001.01









  


  










Input Form





Re[LogIntegral[x + I y]] == Sum[(1/(2^k (k k!))) Cos[k ArcTan[(2 ArcTan[x, y])/Log[x^2 + y^2]]] (1 + (4 ArcTan[x, y]^2)/Log[x^2 + y^2]^2)^(k/2) Log[x^2 + y^2]^k, {k, 1, Infinity}] + (1/2) Log[ArcTan[x, y]^2 + (1/4) Log[x^2 + y^2]^2] + EulerGamma










Standard Form





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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> Re </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> li </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> x </mi> <mo> + </mo> <mrow> <mi> &#8520; </mi> <mo> &#8290; </mo> <mi> y </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <msup> <mn> 2 </mn> <mrow> <mo> - </mo> <mi> k </mi> </mrow> </msup> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> &#8290; </mo> <mi> cos </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mfrac> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <msup> <mrow> <msup> <mi> tan </mi> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mo> ( </mo> <mrow> <mi> x </mi> <mo> , </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mrow> <msup> <mi> log </mi> <mn> 2 </mn> </msup> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mi> k </mi> </msup> <mo> ( </mo> <mrow> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mo> + </mo> <msup> <mi> y </mi> <mn> 2 </mn> </msup> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> + </mo> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <real /> <apply> <ci> LogIntegral </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <imaginaryi /> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <ln /> <apply> <plus /> <apply> <power /> <apply> <arctan /> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <power /> <apply> <ln /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <times /> <ci> k </ci> <apply> <factorial /> <ci> k </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> cos </ci> <apply> <times /> <ci> k </ci> <apply> <arctan /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <arctan /> <ci> x </ci> <ci> y </ci> </apply> <apply> <power /> <apply> <ln /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <power /> <apply> <arctan /> <ci> x </ci> <ci> y </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <ln /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <times /> <ci> k </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <ln /> <apply> <plus /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <ci> k </ci> </apply> </apply> </apply> <eulergamma /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Re", "[", RowBox[List["LogIntegral", "[", RowBox[List["x_", "+", RowBox[List["\[ImaginaryI]", " ", "y_"]]]], "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], " ", RowBox[List["Cos", "[", RowBox[List["k", " ", RowBox[List["ArcTan", "[", FractionBox[RowBox[List["2", " ", RowBox[List["ArcTan", "[", RowBox[List["x", ",", "y"]], "]"]]]], RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], "]"]]], "]"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List["4", " ", SuperscriptBox[RowBox[List["ArcTan", "[", RowBox[List["x", ",", "y"]], "]"]], "2"]]], SuperscriptBox[RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], "]"]], "2"]]]], ")"]], RowBox[List["k", "/", "2"]]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], "]"]], "k"]]], RowBox[List["k", " ", RowBox[List["k", "!"]]]]]]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Log", "[", RowBox[List[SuperscriptBox[RowBox[List["ArcTan", "[", RowBox[List["x", ",", "y"]], "]"]], "2"], "+", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], "]"]], "2"]]]]], "]"]]]], "+", "EulerGamma"]]]]]]










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





2001-10-29