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Sin






Mathematica Notation

Traditional Notation









Elementary Functions > Sin[z] > Series representations > Generalized power series > Expansions at z==Pi/2 > For powers of the function > For symbolical integer power





http://functions.wolfram.com/01.06.06.0062.01









  


  










Input Form





Sin[z]^n \[Proportional] 1 + (1/2^(n + 2)) ((-2^(1 + n)) n + (1 + (-1)^n) Binomial[n, Floor[n/2]] (n - 2 Floor[n/2])^2) (z - Pi/2)^2 + (1/(3 2^(n + 4))) (2^(1 + n) n (-2 + 3 n) - (1 + (-1)^n) Binomial[n, Floor[n/2]] (n - 2 Floor[n/2])^4) (z - Pi/2)^4 + \[Ellipsis] /; (z -> Pi/2) && Element[n, Integers] && n > 0










Standard Form





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MathML Form







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</mo> <mrow> <mo> &#8970; </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> &#8971; </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mo> &#8230; </mo> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mfrac> <mi> &#960; </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <power /> <apply> <sin /> <ci> z </ci> </apply> <ci> n </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <ci> Binomial </ci> <ci> n </ci> <apply> <floor /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 4 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> n </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <ci> n </ci> </apply> <cn type='integer'> -2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <ci> Binomial </ci> <ci> n </ci> <apply> <floor /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <cn type='integer'> 4 </cn> </apply> </apply> <ci> &#8230; </ci> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> z </ci> <apply> <times /> <pi /> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", SuperscriptBox[RowBox[List["Sin", "[", "z_", "]"]], "n_"], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["2", RowBox[List["1", "+", "n"]]]]], " ", "n"]], "+", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], ")"]], "2"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox["\[Pi]", "2"]]], ")"]], "2"]]], SuperscriptBox["2", RowBox[List["n", "+", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["1", "+", "n"]]], " ", "n", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", RowBox[List["3", " ", "n"]]]], ")"]]]], "-", RowBox[List[RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], ")"]], "4"]]]]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", FractionBox["\[Pi]", "2"]]], ")"]], "4"]]], RowBox[List["3", " ", SuperscriptBox["2", RowBox[List["n", "+", "4"]]]]]], "+", "\[Ellipsis]"]], "/;", RowBox[List[RowBox[List["(", RowBox[List["z", "\[Rule]", FractionBox["\[Pi]", "2"]]], ")"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]










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





2007-05-02





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