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variants of this functions
Log






Mathematica Notation

Traditional Notation









Elementary Functions > Log[z] > Series representations > Generalized power series > Expansions of log(1+z) at z==0 > Expansions of log(f(z)) at z==0





http://functions.wolfram.com/01.04.06.0033.01









  


  










Input Form





Log[f[z]] == 2 I Pi Floor[(Pi - Arg[Subscript[c, 0]] - Arg[f[z]/Subscript[c, 0]])/ (2 Pi)] + Log[Subscript[c, 0]] + z Sum[((-1)^k/(k + 1)) (Subscript[c, 1]/Subscript[c, 0])^(k + 1) Subscript[p, k + 1, s - k] z^s, {s, 0, Infinity}, {k, 0, s}] /; f[z] == Sum[Subscript[c, k] z^k, {k, 0, Infinity}] && Subscript[c, 0] != 0 && Subscript[c, 1] != 0 && Subscript[p, j, 0] == 1 && Subscript[p, j, k] == (1/(Subscript[c, 1] k)) Sum[(j m - k + m) Subscript[c, m + 1] Subscript[p, j, k - m], {m, 1, k}] && Element[k, Integers] && k > 0










Standard Form





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MathML Form







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</ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Log", "[", RowBox[List["f", "[", "z_", "]"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", SubscriptBox["c", "0"], "]"]], "-", RowBox[List["Arg", "[", FractionBox[RowBox[List["f", "[", "z", "]"]], SubscriptBox["c", "0"]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["Log", "[", SubscriptBox["c", "0"], "]"]], "+", RowBox[List["z", " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "s"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox[SubscriptBox["c", "1"], SubscriptBox["c", "0"]], ")"]], RowBox[List["k", "+", "1"]]], " ", SubscriptBox["p", RowBox[List[RowBox[List["k", "+", "1"]], ",", RowBox[List["s", "-", "k"]]]]], " ", SuperscriptBox["z", "s"]]], RowBox[List["k", "+", "1"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["f", "[", "z", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["c", "k"], " ", SuperscriptBox["z", "k"]]]]]]], "&&", RowBox[List[SubscriptBox["c", "0"], "\[NotEqual]", "0"]], "&&", RowBox[List[SubscriptBox["c", "1"], "\[NotEqual]", "0"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "&&", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", FractionBox[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "k"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["j", " ", "m"]], "-", "k", "+", "m"]], ")"]], " ", SubscriptBox["c", RowBox[List["m", "+", "1"]]], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]]]], RowBox[List[SubscriptBox["c", "1"], " ", "k"]]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]]]]]]]]]]










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





2007-05-02





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