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variants of this functions
ProductLog






Mathematica Notation

Traditional Notation









Elementary Functions > ProductLog[z] > Series representations > Generalized power series > Expansions at z==-1/e





http://functions.wolfram.com/01.31.06.0005.01









  


  










Input Form





ProductLog[z] == Sum[Subscript[c, k] p^k, {k, 0, Infinity}] /; p == Sqrt[2] Sqrt[1 + E z] && Abs[1 + E z] < 1 && Subscript[c, 0] == -1 && Subscript[c, 1] == 1 && Subscript[a, 0] == 2 && Subscript[a, 1] == -1 && Subscript[a, k] == Sum[Subscript[c, j] Subscript[c, k - j + 1], {j, 2, k - 1}] && Subscript[c, k] == -(Subscript[a, k]/2) + ((k - 1)/(k + 1)) ((1/4) Subscript[a, k - 2] + (1/2) Subscript[c, k - 2]) - Subscript[c, k - 1]/(k + 1)










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ProductLog", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["c", "k"], " ", SuperscriptBox["p", "k"]]]]], "/;", RowBox[List[RowBox[List["p", "\[Equal]", RowBox[List[SqrtBox["2"], " ", SqrtBox[RowBox[List["1", "+", RowBox[List["\[ExponentialE]", " ", "z"]]]]]]]]], "&&", RowBox[List[RowBox[List["Abs", "[", RowBox[List["1", "+", RowBox[List["\[ExponentialE]", " ", "z"]]]], "]"]], "<", "1"]], "&&", RowBox[List[SubscriptBox["c", "0"], "\[Equal]", RowBox[List["-", "1"]]]], "&&", RowBox[List[SubscriptBox["c", "1"], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["a", "0"], "\[Equal]", "2"]], "&&", RowBox[List[SubscriptBox["a", "1"], "\[Equal]", RowBox[List["-", "1"]]]], "&&", RowBox[List[SubscriptBox["a", "k"], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "2"]], RowBox[List["k", "-", "1"]]], RowBox[List[SubscriptBox["c", "j"], " ", SubscriptBox["c", RowBox[List["k", "-", "j", "+", "1"]]]]]]]]], "&&", RowBox[List[SubscriptBox["c", "k"], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["a", "k"], "2"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]], " ", RowBox[List["(", RowBox[List[FractionBox[SubscriptBox["a", RowBox[List["k", "-", "2"]]], "4"], "+", FractionBox[SubscriptBox["c", RowBox[List["k", "-", "2"]]], "2"]]], ")"]]]], RowBox[List["k", "+", "1"]]], "-", FractionBox[SubscriptBox["c", RowBox[List["k", "-", "1"]]], RowBox[List["k", "+", "1"]]]]]]]]]]]]]]]










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





2001-10-29





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