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LogIntegral






Mathematica Notation

Traditional Notation









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





http://functions.wolfram.com/06.36.19.0002.01









  


  










Input Form





Re[LogIntegral[x + I y]] == Sum[(((-1)^j 2^(2 j - k))/(k k!)) Binomial[k, 2 j] ArcTan[x, y]^(2 j) Log[x^2 + y^2]^(k - 2 j), {k, 1, Infinity}, {j, 0, Floor[k/2]}] + (1/2) Log[ArcTan[x, y]^2 + (1/4) Log[x^2 + y^2]^2] + EulerGamma










Standard Form





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MathML Form







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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]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["Floor", "[", FractionBox["k", "2"], "]"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["2", " ", "j"]], "-", "k"]]]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", RowBox[List["2", " ", "j"]]]], "]"]], " ", SuperscriptBox[RowBox[List["ArcTan", "[", RowBox[List["x", ",", "y"]], "]"]], RowBox[List["2", " ", "j"]]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List[SuperscriptBox["x", "2"], "+", SuperscriptBox["y", "2"]]], "]"]], RowBox[List["k", "-", RowBox[List["2", " ", "j"]]]]]]], 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





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