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LogIntegral






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > LogIntegral[z] > Series representations > Generalized power series > Expansions on branch cuts > For the function itself





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







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</apply> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <factorial /> <ci> j </ci> </apply> <apply> <ci> StirlingS1 </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> <ci> j </ci> </apply> <apply> <power /> <apply> <ln /> <ci> x </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> x </ci> <reals /> </apply> <apply> <lt /> <ci> x </ci> <cn type='integer'> 0 </cn> </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[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





© 1998- Wolfram Research, Inc.