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Power




Mathematica Notation

Traditional Notation









Elementary Functions > Power[z,a] > Series representations > Generalized power series > Expansions at generic point z==z0 > Expansions of f(z)a at z==z0





http://functions.wolfram.com/01.02.06.0035.01









  


  








Input Form





f[z]^a \[Proportional] E^(2 I a Pi Floor[1/2 - Arg[f[Subscript[z, 0]]]/(2 Pi) - (1/(2 Pi)) Arg[f[z]/f[Subscript[z, 0]]]]) f[Subscript[z, 0]]^a (1 + ((a Derivative[1][f][Subscript[z, 0]])/f[Subscript[z, 0]]) (z - Subscript[z, 0]) + (a/(2 f[Subscript[z, 0]]^2)) ((-1 + a) Derivative[1][f][Subscript[z, 0]]^2 + f[Subscript[z, 0]] Derivative[2][f][Subscript[z, 0]]) (z - Subscript[z, 0])^2 + \[Ellipsis]) /; (z -> Subscript[z, 0])








Standard Form





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MathML Form







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Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["f", "[", "z_", "]"]], "a_"], "]"]], "\[RuleDelayed]", RowBox[List[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", "[", SubscriptBox["zz", "0"], "]"]], "]"]], RowBox[List["2", " ", "\[Pi]"]]], "-", FractionBox[RowBox[List["Arg", "[", FractionBox[RowBox[List["f", "[", "z", "]"]], RowBox[List["f", "[", SubscriptBox["zz", "0"], "]"]]], "]"]], RowBox[List["2", " ", "\[Pi]"]]]]], "]"]]]]], " ", SuperscriptBox[RowBox[List["f", "[", SubscriptBox["zz", "0"], "]"]], "a"], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["a", " ", RowBox[List[SuperscriptBox["f", "\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["zz", "0"], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], RowBox[List["f", "[", SubscriptBox["zz", "0"], "]"]]], "+", FractionBox[RowBox[List["a", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "a"]], ")"]], " ", SuperscriptBox[RowBox[List[SuperscriptBox["f", "\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["zz", "0"], "]"]], "2"]]], "+", RowBox[List[RowBox[List["f", "[", SubscriptBox["zz", "0"], "]"]], " ", RowBox[List[SuperscriptBox["f", "\[Prime]\[Prime]", Rule[MultilineFunction, None]], "[", SubscriptBox["zz", "0"], "]"]]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], "2"]]], RowBox[List["2", " ", SuperscriptBox[RowBox[List["f", "[", SubscriptBox["zz", "0"], "]"]], "2"]]]], "+", "\[Ellipsis]"]], ")"]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", SubscriptBox["zz", "0"]]], ")"]]]]]]]]


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





2007-05-02