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 http://functions.wolfram.com/06.36.06.0021.01

 Input Form

 LogIntegral[z] == (-Floor[Arg[z - x]/(2 Pi)]) ExpIntegralEi[Log[-x] - I Pi] + (Floor[Arg[z - x]/(2 Pi)] + 1) LogIntegral[x] - 2 I Pi Floor[Arg[z - x]/(2 Pi)] Sum[(((z - x)^k x^(1 - k))/k!) Sum[(((-1)^j j! StirlingS1[k - 1, j])/(Log[-x]^2 + Pi^2)^(j + 1)) Sum[Binomial[j + 1, 2 h + 1] Log[-x]^(j - 2 h) (I Pi)^(2 h), {h, 0, Floor[(j + 1)/2]}], {j, 0, k - 1}], {k, 1, n}] + Sum[(1/k!) (z - x)^k x^(1 - k) Sum[(-1)^j j! StirlingS1[k - 1, j] Log[x]^(-j - 1), {j, 0, k - 1}], {k, 1, n}] /; Element[x, Reals] && x < 0

 Standard Form

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 MathML Form

 li ( z ) - 2 π arg ( z - x ) 2 π k = 1 ( z - x ) k x 1 - k k ! j = 0 k - 1 ( - 1 ) j j ! S TagBox["S", StirlingS1] k - 1 ( j ) ( log 2 ( - x ) + π 2 ) j + 1 h = 0 j + 1 2 ( j + 1 2 h + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["j", "+", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "h"]], "+", "1"]], Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] log j - 2 h ( - x ) ( π ) 2 h - arg ( z - x ) 2 π Ei ( log ( - x ) - π ) + ( arg ( z - x ) 2 π + 1 ) li ( x ) + k = 1 ( z - x ) k x 1 - k k ! j = 0 k - 1 ( - 1 ) j j ! S TagBox["S", StirlingS1] k - 1 ( j ) log - j - 1 ( x ) /; x TagBox["\[DoubleStruckCapitalR]", Function[List[], Reals]] x < 0 Condition LogIntegral z -2 z -1 x 2 -1 k 1 z -1 x k x 1 -1 k k -1 j 0 k -1 -1 j j StirlingS1 k -1 j -1 x 2 2 j 1 -1 h 0 j 1 2 -1 Binomial j 1 2 h 1 -1 x j -1 2 h 2 h -1 z -1 x 2 -1 ExpIntegralEi -1 x -1 z -1 x 2 -1 1 LogIntegral x k 1 z -1 x k x 1 -1 k k -1 j 0 k -1 -1 j j StirlingS1 k -1 j x -1 j -1 x x 0 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LogIntegral", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List[RowBox[List["Log", "[", RowBox[List["-", "x"]], "]"]], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]]]], "]"]]]], "+", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], "+", "1"]], ")"]], " ", RowBox[List["LogIntegral", "[", "x", "]"]]]], "-", RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["Arg", "[", RowBox[List["z", "-", "x"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "]"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"], " ", SuperscriptBox["x", RowBox[List["1", "-", "k"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["j", "!"]], " ", RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["k", "-", "1"]], ",", "j"]], "]"]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["j", "+", "1"]], "2"], "]"]]], RowBox[List[RowBox[List["Binomial", "[", RowBox[List[RowBox[List["j", "+", "1"]], ",", RowBox[List[RowBox[List["2", " ", "h"]], "+", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", "x"]], "]"]], RowBox[List["j", "-", RowBox[List["2", " ", "h"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["\[ImaginaryI]", " ", "\[Pi]"]], ")"]], RowBox[List["2", " ", "h"]]]]]]]]], SuperscriptBox[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["Log", "[", RowBox[List["-", "x"]], "]"]], "2"], "+", SuperscriptBox["\[Pi]", "2"]]], ")"]], RowBox[List["j", "+", "1"]]]]]]]], RowBox[List["k", "!"]]]]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", "x"]], ")"]], "k"], " ", SuperscriptBox["x", RowBox[List["1", "-", "k"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["k", "-", "1"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["j", "!"]], " ", RowBox[List["StirlingS1", "[", RowBox[List[RowBox[List["k", "-", "1"]], ",", "j"]], "]"]], " ", SuperscriptBox[RowBox[List["Log", "[", "x", "]"]], RowBox[List[RowBox[List["-", "j"]], "-", "1"]]]]]]]]], RowBox[List["k", "!"]]]]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["x", "<", "0"]]]]]]]]]]

 Contributed by

 Pavlyk 0. (2006)

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

 2007-05-02