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variants of this functions
ProductLog






Mathematica Notation

Traditional Notation









Elementary Functions > ProductLog[k,z] > Series representations > Asymptotic series expansions





http://functions.wolfram.com/01.32.06.0002.01









  


  










Input Form





ProductLog[k, z] == w - \[Zeta] - Sum[Sum[(((-1)^n StirlingS1[n, 1 - m + n])/m!) \[Zeta]^m, {m, 1, n}]/w^n, {n, 1, Infinity}] /; (z -> 0 | Infinity) && w == Log[z] + 2 I k Pi && \[Zeta] == Log[w] && !(k == 0 && Abs[z] < E^2) && !(k == -1 && IntervalMemberQ[Interval[{-(1/E), 0}], z])










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ProductLog", "[", RowBox[List["k_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["w", "-", "\[Zeta]", "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "1"]], "\[Infinity]"], RowBox[List[SuperscriptBox["w", RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["StirlingS1", "[", RowBox[List["n", ",", RowBox[List["1", "-", "m", "+", "n"]]]], "]"]]]], ")"]], " ", SuperscriptBox["\[Zeta]", "m"]]], RowBox[List["m", "!"]]]]]]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", RowBox[List["0", "|", "\[Infinity]"]]]], ")"]], "&&", RowBox[List["w", "\[Equal]", RowBox[List[RowBox[List["Log", "[", "z", "]"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "k", " ", "\[Pi]"]]]]]], "&&", RowBox[List["\[Zeta]", "\[Equal]", RowBox[List["Log", "[", "w", "]"]]]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["k", "\[Equal]", "0"]], "&&", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "<", SuperscriptBox["\[ExponentialE]", "2"]]]]], ")"]]]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["k", "\[Equal]", RowBox[List["-", "1"]]]], "&&", RowBox[List["IntervalMemberQ", "[", RowBox[List[RowBox[List["Interval", "[", RowBox[List["{", RowBox[List[RowBox[List["-", FractionBox["1", "\[ExponentialE]"]]], ",", "0"]], "}"]], "]"]], ",", "z"]], "]"]]]], ")"]]]]]]]]]]]]










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





2001-10-29