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 Log

 http://functions.wolfram.com/01.04.20.0025.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "\[Alpha]_"]], "}"]]]]], RowBox[List["Log", "[", RowBox[List["f", "[", "z_", "]"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "\[ImaginaryI]", " ", "\[Pi]", " ", RowBox[List["Floor", "[", FractionBox[RowBox[List["\[Pi]", "-", RowBox[List["Arg", "[", RowBox[List["f", "[", "0", "]"]], "]"]], "-", RowBox[List["Arg", "[", FractionBox[RowBox[List["f", "[", "z", "]"]], RowBox[List["f", "[", "0", "]"]]], "]"]]]], RowBox[List["2", " ", "\[Pi]"]]], "]"]]]], "+", RowBox[List["Log", "[", RowBox[List["f", "[", "0", "]"]], "]"]]]], ")"]], " ", SuperscriptBox["z", RowBox[List["-", "\[Alpha]"]]]]], RowBox[List["Gamma", "[", RowBox[List["1", "-", "\[Alpha]"]], "]"]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["s", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], FractionBox["s", "u"]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox[RowBox[List["(", FractionBox[RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", "u", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "0", "]"]], RowBox[List[RowBox[List["f", "[", "0", "]"]], " ", RowBox[List["u", "!"]]]]], ")"]], RowBox[List["k", "+", "1"]]], " ", SubscriptBox["p", RowBox[List[RowBox[List["k", "+", "1"]], ",", RowBox[List["s", "-", RowBox[List["u", " ", "k"]]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", "u", "+", "1"]], "]"]], " ", SuperscriptBox["z", RowBox[List["s", "+", "u", "-", "\[Alpha]"]]]]], RowBox[List[RowBox[List["(", RowBox[List["k", "+", "1"]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List["s", "+", "u", "-", "\[Alpha]", "+", "1"]], "]"]]]]]]]]]]], "/;", RowBox[List[RowBox[List[RowBox[List["f", "[", "0", "]"]], "\[NotEqual]", "0"]], "&&", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "0", "]"]], "\[Equal]", "0"]], "/;", RowBox[List["1", "\[LessEqual]", "k", "\[LessEqual]", RowBox[List["u", "-", "1"]]]]]], ")"]], "&&", RowBox[List[RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", "u", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "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[RowBox[List["u", "!"]], " ", 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", "+", "u"]], ")"]], Derivative], Rule[MultilineFunction, None]], "[", "0", "]"]], " ", SubscriptBox["p", RowBox[List["j", ",", RowBox[List["k", "-", "m"]]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["m", "+", "u"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["m", "+", "1"]], ")"]], "!"]]]]]]]]], RowBox[List[RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", "u", ")"]], Derivative], Rule[MultilineFunction, None]], "[", "0", "]"]], " ", "k"]]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]]]]]]]]]]

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

 2007-05-02