Sin[n z]/Sin[z] == Sum[Product[If[k == j, 1, Sin[z + Subscript[\[Theta], j] - Subscript[\[Theta], k]]/Sin[Subscript[\[Theta], j] - Subscript[\[Theta], k]]], {k, 1, n}], {j, 0, n}] /; (Element[n, Integers] && n >= 0) && Subscript[\[Theta], j] != Subscript[\[Theta], k]

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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> n </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mfrac> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <munder> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mrow> <mi> k </mi> <mo> &#8800; </mo> <mi> j </mi> </mrow> </munder> <mi> n </mi> </munderover> <mfrac> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <msub> <mi> &#952; </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> &#952; </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> &#952; </mi> <mi> j </mi> </msub> <mo> - </mo> <msub> <mi> &#952; </mi> <mi> k </mi> </msub> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> &#952; </mi> <mi> j </mi> </msub> <mo> &#8800; </mo> <msub> <mi> &#952; </mi> <mi> k </mi> </msub> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> z </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <ms> z </ms> <ms> ) </ms> </list> </apply> </apply> <ms> &#10869; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 0 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8719; </ms> <apply> <ci> UnderscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> &#8800; </ms> <ms> j </ms> </list> </apply> </apply> <ms> n </ms> </apply> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> z </ms> <ms> + </ms> <apply> <ci> SubscriptBox </ci> <ms> &#952; </ms> <ms> j </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> &#952; </ms> <ms> k </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#952; </ms> <ms> j </ms> </apply> <ms> - </ms> <apply> <ci> SubscriptBox </ci> <ms> &#952; </ms> <ms> k </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </apply> </list> </apply> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> &#8712; </ms> <ms> &#8469; </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> &#952; </ms> <ms> j </ms> </apply> <ms> &#8800; </ms> <apply> <ci> SubscriptBox </ci> <ms> &#952; </ms> <ms> k </ms> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox[RowBox[List["Sin", "[", RowBox[List["n_", " ", "z_"]], "]"]], RowBox[List["Sin", "[", "z_", "]"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], RowBox[List["If", "[", RowBox[List[RowBox[List["k", "\[Equal]", "j"]], ",", "1", ",", FractionBox[RowBox[List["Sin", "[", RowBox[List["z", "+", SubscriptBox["\[Theta]", "j"], "-", SubscriptBox["\[Theta]", "k"]]], "]"]], RowBox[List["Sin", "[", RowBox[List[SubscriptBox["\[Theta]", "j"], "-", SubscriptBox["\[Theta]", "k"]]], "]"]]]]], "]"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]], ")"]], "&&", RowBox[List[SubscriptBox["\[Theta]", "j"], "\[NotEqual]", SubscriptBox["\[Theta]", "k"]]]]]]]]]]]

2001-10-29