Sine: Transformations (formula 01.06.16.0113)

Sin

$Mathematica Notation$

Elementary Functions

Sin[z]

Transformations

Products, sums, and powers of the direct function

Products involving the direct function

http://functions.wolfram.com/01.06.16.0113.01

Input Form

Product[Sin[Subscript[z, k]], {k, 1, n}] == Sum[Cos[Sum[Subscript[k, j] (Subscript[z, j] + Pi/2), {j, 1, n}]], {Subscript[k, 1], -1, 1, 2}, {Subscript[k, 2], -1, 1, 2}, â€¦, {Subscript[k, n], -1, 1, 2}]/(-2)^n

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], RowBox[List["Sin", "[", SubscriptBox["z", "k"], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "2"]], ")"]], RowBox[List["-", "n"]]], RowBox[List[UnderoverscriptBox["\[Sum]", UnderscriptBox[RowBox[List[SubscriptBox["k", "1"], "=", RowBox[List["-", "1"]]]], RowBox[List[RowBox[List["\[CapitalDelta]", " ", SubscriptBox["k", "1"]]], "=", "2"]]], "1"], RowBox[List[UnderoverscriptBox["\[Sum]", UnderscriptBox[RowBox[List[SubscriptBox["k", "2"], "=", RowBox[List["-", "1"]]]], RowBox[List[RowBox[List["\[CapitalDelta]", " ", SubscriptBox["k", "2"]]], "=", "2"]]], "1"], RowBox[List["\[Ellipsis]", RowBox[List[UnderoverscriptBox["\[Sum]", UnderscriptBox[RowBox[List[SubscriptBox["k", "n"], "=", RowBox[List["-", "1"]]]], RowBox[List[RowBox[List["\[CapitalDelta]", " ", SubscriptBox["k", "n"]]], "=", "2"]]], "1"], RowBox[List["Cos", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[SubscriptBox["k", "j"], RowBox[List["(", RowBox[List[SubscriptBox["z", "j"], "+", FractionBox["\[Pi]", "2"]]], ")"]]]]]], "]"]]]]]]]]]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mi> sin </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <munder> <mrow> <msub> <mi> k </mi> <mn> 1 </mn> </msub> <mo> = </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mrow> <mrow> <mi> Δ </mi> <mo> ⁢ </mo> <msub> <mi> k </mi> <mn> 1 </mn> </msub> </mrow> <mo> = </mo> <mn> 2 </mn> </mrow> </munder> <mn> 1 </mn> </munderover> <mrow> <munderover> <mo> ∑ </mo> <munder> <mrow> <msub> <mi> k </mi> <mn> 2 </mn> </msub> <mo> = </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mrow> <mrow> <mi> Δ </mi> <mo> ⁢ </mo> <msub> <mi> k </mi> <mn> 2 </mn> </msub> </mrow> <mo> = </mo> <mn> 2 </mn> </mrow> </munder> <mn> 1 </mn> </munderover> <mrow> <mo> … </mo> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <munder> <mrow> <msub> <mi> k </mi> <mi> n </mi> </msub> <mo> = </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> </mrow> <mrow> <mrow> <mi> Δ </mi> <mo> ⁢ </mo> <msub> <mi> k </mi> <mi> n </mi> </msub> </mrow> <mo> = </mo> <mn> 2 </mn> </mrow> </munder> <mn> 1 </mn> </munderover> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msub> <mi> k </mi> <mi> j </mi> </msub> <mo> ( </mo> <mrow> <msub> <mi> z </mi> <mi> j </mi> </msub> <mo> + </mo> <mfrac> <mi> π </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∏ </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> sin </ms> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> k </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ⩵ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 2 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> n </ms> </list> </apply> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∑ </ms> <apply> <ci> UnderscriptBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> <ms> = </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Δ </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 1 </ms> </apply> </list> </apply> <ms> = </ms> <ms> 2 </ms> </list> </apply> </apply> <ms> 1 </ms> </apply> <apply> <ci> ErrorBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∑ </ms> <apply> <ci> UnderscriptBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> <ms> = </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Δ </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> = </ms> <ms> 2 </ms> </list> </apply> </apply> <ms> 1 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> … </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∑ </ms> <apply> <ci> UnderscriptBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> n </ms> </apply> <ms> = </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> Δ </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> n </ms> </apply> </list> </apply> <ms> = </ms> <ms> 2 </ms> </list> </apply> </apply> <ms> 1 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> cos </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∑ </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <ms> 1 </ms> </list> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> j </ms> </apply> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> j </ms> </apply> <ms> + </ms> <apply> <ci> FractionBox </ci> <ms> π </ms> <ms> 2 </ms> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>

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

2002-12-18