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 Power

 http://functions.wolfram.com/01.02.20.0028.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "\[Alpha]"]], "}"]]], SuperscriptBox[RowBox[List["f", "[", "z", "]"]], "a"]]], "\[Equal]", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", "\[ImaginaryI]", " ", "a", " ", "\[Pi]", " ", RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], "-", FractionBox[RowBox[List["Arg", "[", RowBox[List["f", "[", "0", "]"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "-", RowBox[List[FractionBox["1", RowBox[List["2", " ", "\[Pi]"]]], RowBox[List["Arg", "[", FractionBox[RowBox[List["f", "[", "z", "]"]], RowBox[List["f", "[", "0", "]"]]], "]"]]]]]], "]"]]]]], " ", FractionBox["a", RowBox[List["Gamma", "[", RowBox[List["1", "-", "a"]], "]"]]], " ", SuperscriptBox[RowBox[List["f", "[", "0", "]"]], "a"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List["Gamma", "[", RowBox[List["1", "+", "k", "-", "a"]], "]"]], RowBox[List["Gamma", "[", RowBox[List["1", "+", "k", "-", "\[Alpha]"]], "]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]]]], RowBox[List["a", "-", "j"]]], SubscriptBox["p", RowBox[List["j", ",", "k"]]], SuperscriptBox["z", RowBox[List["k", "-", "\[Alpha]"]]]]]]]]]]]]]]], "/;", RowBox[List[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["f", "[", "0", "]"]], "k"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["m", "=", "1"]], "k"], RowBox[List[FractionBox[RowBox[List[RowBox[List["j", " ", "m"]], "-", "k", "+", "m"]], RowBox[List["m", "!"]]], RowBox[List[SuperscriptBox["f", TagBox[RowBox[List["(", "m", ")"]], 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"]], "\[And]", RowBox[List[RowBox[List["f", "[", "0", "]"]], "\[NotEqual]", "0"]]]]]]]]

 MathML Form

 α f ( z ) a z α a Γ ( 1 - a ) 2 a π 1 2 - arg ( f ( z ) f ( 0 ) ) 2 π - arg ( f ( 0 ) ) 2 π f ( 0 ) a k = 0 Γ ( k - a + 1 ) Γ ( k - α + 1 ) j = 0 k ( - 1 ) j a - j ( k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] p j , k z k - α /; p j , 0 1 p j , k 1 f ( 0 ) k m = 1 k j m + m - k m ! f ( m ) TagBox[RowBox[List["(", "m", ")"]], Derivative] ( 0 ) p j , k - m k + f ( 0 ) 0 Condition z α f z a a Gamma 1 -1 a -1 2 a 1 2 -1 f z f 0 -1 2 -1 -1 f 0 2 -1 f 0 a k 0 Gamma k -1 a 1 Gamma k -1 α 1 -1 j 0 k -1 j a -1 j -1 Binomial k j Subscript p j k z k -1 α Subscript p j 0 1 Subscript p j k 1 f 0 k -1 m 1 k j m m -1 k m -1 D f 0 0 m Subscript p j k -1 m k SuperPlus f 0 0 [/itex]

 Rule Form

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

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

 2007-05-02