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Sin






Mathematica Notation

Traditional Notation









Elementary Functions > Sin[z] > Series representations > Other series representations





http://functions.wolfram.com/01.06.06.0029.01









  


  










Input Form





Sin[z] == z (Sum[(-1)^(-n + k) Binomial[2 k, -n + k] Binomial[1 - n + k - z/(c Pi), 1 + 2 k], {k, n, Infinity}] - Sum[(-1)^(-n + k) Binomial[2 k, 1 - n + k] Binomial[n + k + z/(c Pi), 1 + 2 k], {k, -1 + n, Infinity}]) /; Element[n, Integers]










Standard Form





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MathML Form







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</mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <sin /> <ci> z </ci> </apply> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <ci> n </ci> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <ci> Binomial </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <ci> c </ci> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> 1 </cn> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <apply> <plus /> <ci> n </ci> <cn type='integer'> -1 </cn> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <ci> Binomial </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Binomial </ci> <apply> <plus /> <ci> k </ci> <ci> n </ci> <apply> <times /> <ci> z </ci> <apply> <power /> <apply> <times /> <ci> c </ci> <pi /> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Sin", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "n"]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "n"]], "+", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["2", " ", "k"]], ",", RowBox[List[RowBox[List["-", "n"]], "+", "k"]]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["1", "-", "n", "+", "k", "-", FractionBox["z", RowBox[List["c", " ", "\[Pi]"]]]]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "]"]]]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List[RowBox[List["-", "1"]], "+", "n"]]]], "\[Infinity]"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "n"]], "+", "k"]]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["2", " ", "k"]], ",", RowBox[List["1", "-", "n", "+", "k"]]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["n", "+", "k", "+", FractionBox["z", RowBox[List["c", " ", "\[Pi]"]]]]], ",", RowBox[List["1", "+", RowBox[List["2", " ", "k"]]]]]], "]"]]]]]]]], ")"]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]]]










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





2002-12-18





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