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 Log

 http://functions.wolfram.com/01.04.06.0020.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Log", "[", RowBox[List["f", "[", "z", "]"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", RowBox[List["f", "[", SubscriptBox["z", "0"], "]"]], "]"]], "-", RowBox[List["Arg", "[", FractionBox[RowBox[List["f", "[", "z", "]"]], RowBox[List["f", "[", SubscriptBox["z", "0"], "]"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["f", "[", SubscriptBox["z", "0"], "]"]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "s"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[SuperscriptBox["f", "\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["z", "0"], "]"]], RowBox[List["f", "[", SubscriptBox["z", "0"], "]"]]], ")"]], RowBox[List["k", "+", "1"]]]]], RowBox[List["k", "+", "1"]]], SubscriptBox["p", RowBox[List[RowBox[List["k", "+", "1"]], ",", RowBox[List["s", "-", "k"]]]]], SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], RowBox[List["s", "+", "1"]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["f", "[", SubscriptBox["z", "0"], "]"]], "\[NotEqual]", "0"]], "\[And]", RowBox[List[RowBox[List[SuperscriptBox["f", "\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["z", "0"], "]"]], "\[NotEqual]", "0"]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "0"]]], "\[Equal]", "1"]], "\[And]", RowBox[List[SubscriptBox["p", RowBox[List["j", ",", "k"]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List[RowBox[List[SuperscriptBox["f", "\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["z", "0"], "]"]], "k"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "k"], RowBox[List[FractionBox[RowBox[List[RowBox[List["j", " ", "m"]], "-", "k", "+", "m"]], RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]]], RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", SubscriptBox["z", "0"], "]"]], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]]]]]]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", ">", "0"]]]]]]]]

 MathML Form

 log ( f ( z ) ) 2 π π - arg ( f ( z 0 ) ) - arg ( f ( z ) f ( z 0 ) ) 2 π + log ( f ( z 0 ) ) + s = 0 k = 0 s ( - 1 ) k k + 1 ( f ( z 0 ) f ( z 0 ) ) k + 1 p k + 1 , s - k ( z - z 0 ) s + 1 /; f ( z 0 ) 0 f ( z 0 ) 0 p j , 0 1 p j , k 1 f ( z 0 ) k m = 1 k j m + m - k ( m + 1 ) ! f ( m + 1 ) TagBox[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], Derivative] ( z 0 ) p j , k - m k + Condition f z 2 -1 f Subscript z 0 -1 f z f Subscript z 0 -1 2 -1 f Subscript z 0 k 0 s s 0 -1 k k 1 -1 D f Subscript z 0 Subscript z 0 f Subscript z 0 -1 k 1 Subscript p k 1 s -1 k z -1 Subscript z 0 s 1 f Subscript z 0 0 D f Subscript z 0 Subscript z 0 0 Subscript p j 0 1 Subscript p j k 1 D f Subscript z 0 Subscript z 0 k -1 m 1 k j m m -1 k m 1 -1 D f Subscript z 0 Subscript z 0 m 1 Subscript p j k -1 m k SuperPlus [/itex]

 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", "[", RowBox[List["f", "[", SubscriptBox["zz", "0"], "]"]], "]"]], "-", RowBox[List["Arg", "[", FractionBox[RowBox[List["f", "[", "z", "]"]], RowBox[List["f", "[", SubscriptBox["zz", "0"], "]"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["f", "[", SubscriptBox["zz", "0"], "]"]], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "s"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[SuperscriptBox["f", "\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["zz", "0"], "]"]], RowBox[List["f", "[", SubscriptBox["zz", "0"], "]"]]], ")"]], RowBox[List["k", "+", "1"]]]]], ")"]], " ", SubscriptBox["p", RowBox[List[RowBox[List["k", "+", "1"]], ",", RowBox[List["s", "-", "k"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], RowBox[List["s", "+", "1"]]]]], RowBox[List["k", "+", "1"]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["f", "[", SubscriptBox["zz", "0"], "]"]], "\[NotEqual]", "0"]], "&&", RowBox[List[RowBox[List[SuperscriptBox["f", "\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["zz", "0"], "]"]], "\[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"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["j", " ", "m"]], "-", "k", "+", "m"]], ")"]], " ", RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", SubscriptBox["zz", "0"], "]"]], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]], RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]]]]], RowBox[List[RowBox[List[SuperscriptBox["f", "\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["zz", "0"], "]"]], " ", "k"]]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]]]]]]]]]]

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

 2007-05-02