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 Sin

 http://functions.wolfram.com/01.06.06.0046.01

 Input Form

 Sin[z]^n \[Proportional] z^n (1 - ((2^(1 - n)/(2 + n)!) Sum[(-1)^k Binomial[n, k] (n - 2 k)^(n + 2), {k, 0, Floor[(n - 1)/2]}]) z^2 + ((2^(1 - n)/(n + 4)!) Sum[(-1)^k Binomial[n, k] (n - 2 k)^(n + 4), {k, 0, Floor[(n - 1)/2]}]) z^4 + O[z^6]) /; Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "n"], "\[Proportional]", RowBox[List[SuperscriptBox["z", "n"], RowBox[List["(", " ", RowBox[List["1", "-", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["2", RowBox[List["1", "-", "n"]]], RowBox[List[RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], "!"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List["n", "+", "2"]]]]]]]]], ")"]], SuperscriptBox["z", "2"]]], "+", RowBox[List[RowBox[List["(", RowBox[List[FractionBox[SuperscriptBox["2", RowBox[List["1", "-", "n"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "4"]], ")"]], "!"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List["n", "+", "4"]]]]]]]]], ")"]], SuperscriptBox["z", "4"]]], "+", RowBox[List["O", "[", SuperscriptBox["z", "6"], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 sin n ( z ) z n ( 1 - ( 2 1 - n ( n + 2 ) ! k = 0 n - 1 2 ( - 1 ) k ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n - 2 k ) n + 2 ) z 2 + ( 2 1 - n ( n + 4 ) ! k = 0 n - 1 2 ( - 1 ) k ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( n - 2 k ) n + 4 ) z 4 + O ( z 6 ) ) /; n + Condition Proportional z n z n 1 -1 2 1 -1 n n 2 -1 k 0 n -1 2 -1 -1 k Binomial n k n -1 2 k n 2 z 2 2 1 -1 n n 4 -1 k 0 n -1 2 -1 -1 k Binomial n k n -1 2 k n 4 z 4 O z 6 n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["Sin", "[", "z_", "]"]], "n_"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["z", "n"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List["n", "+", "2"]]]]]]]]], ")"]], " ", SuperscriptBox["z", "2"]]], RowBox[List[RowBox[List["(", RowBox[List["2", "+", "n"]], ")"]], "!"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["1", "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["n", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", "k"]]]], ")"]], RowBox[List["n", "+", "4"]]]]]]]]], ")"]], " ", SuperscriptBox["z", "4"]]], RowBox[List[RowBox[List["(", RowBox[List["n", "+", "4"]], ")"]], "!"]]], "+", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], "6"]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2007-05-02