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variants of this functions
ProductLog






Mathematica Notation

Traditional Notation









Elementary Functions > ProductLog[k,z] > Differentiation > Symbolic differentiation





http://functions.wolfram.com/01.32.20.0003.01









  


  










Input Form





D[ProductLog[k, z], {z, n}] == Boole[n == 0, ProductLog[k, z]] + ((n! ProductLog[k, z]^n)/(z^n (1 + ProductLog[k, z])^(2 n - 1))) Sum[(((-1)^(n + q - j - 1) (l + n)^(j + l + n - 1) Binomial[l + n, n] Pochhammer[-2 n, q - j - l - m])/(j! (l + n)! (q - j - l - m)!)) ProductLog[k, z]^q, {q, 0, n}, {m, 0, q}, {l, 0, q - m}, {j, 0, q - l - m}] /; Element[n, Integers] && n > 0










Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["ProductLog", "[", RowBox[List["k_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Boole", "[", RowBox[List[RowBox[List["n", "\[Equal]", "0"]], ",", RowBox[List["ProductLog", "[", RowBox[List["k", ",", "z"]], "]"]]]], "]"]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["n", "!"]], " ", SuperscriptBox[RowBox[List["ProductLog", "[", RowBox[List["k", ",", "z"]], "]"]], "n"]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "0"]], "q"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["l", "=", "0"]], RowBox[List["q", "-", "m"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List["q", "-", "l", "-", "m"]]], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["n", "+", "q", "-", "j", "-", "1"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["l", "+", "n"]], ")"]], RowBox[List["j", "+", "l", "+", "n", "-", "1"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["l", "+", "n"]], ",", "n"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "n"]], ",", RowBox[List["q", "-", "j", "-", "l", "-", "m"]]]], "]"]]]], ")"]], " ", SuperscriptBox[RowBox[List["ProductLog", "[", RowBox[List["k", ",", "z"]], "]"]], "q"]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["l", "+", "n"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["q", "-", "j", "-", "l", "-", "m"]], ")"]], "!"]]]]]]]]]]]]]]], RowBox[List[SuperscriptBox["z", "n"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["1", "+", RowBox[List["ProductLog", "[", RowBox[List["k", ",", "z"]], "]"]]]], ")"]], RowBox[List[RowBox[List["2", " ", "n"]], "-", "1"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










Contributed by





Eric Weisstein and Oleg Marichev










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





2007-05-02





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