Cosine: Transformations (formula 01.07.16.0113)

Cos

$Mathematica Notation$

Elementary Functions

Cos[z]

Transformations

Products, sums, and powers of the direct function

Products involving the direct function

http://functions.wolfram.com/01.07.16.0113.01

Input Form

Product[Cos[Subscript[z, k]], {k, 1, m}]* Product[Sin[Subscript[z, k]], {k, m + 1, n}] == 2^-n (-1)^Mod[(n - m)/2, 2]/2^n * Sum[(-1)^Sum[(2 Subscript[k, j] + 2)/4, {j, m + 1, n}]* Cos[Sum[Subscript[k, j] Subscript[z, j], {j, 1, n}]], {Subscript[k, 1], -1, 1, 2}, {Subscript[k, 2], -1, 1, 2}, â€¦, {Subscript[k, n], -1, 1, 2}] /; n âˆˆ Integers && m âˆˆ Integers && n >= 0 && m >= 0 && m <= n && (n - m)/2 âˆˆ Integers

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "m"], RowBox[List["Cos", "[", SubscriptBox["z", "k"], "]"]]]], ")"]], RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", RowBox[List["m", "+", "1"]]]], "n"], RowBox[List["Sin", "[", SubscriptBox["z", "k"], "]"]]]], ")"]]]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Mod", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m"]], ")"]], "/", "2"]], ",", "2"]], "]"]]], SuperscriptBox["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[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", RowBox[List["m", "+", "1"]]]], "n"], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", SubscriptBox["k", "j"]]], "+", "2"]], ")"]], "/", "4"]]]]], RowBox[List["Cos", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "1"]], "n"], RowBox[List[SubscriptBox["k", "j"], SubscriptBox["z", "j"]]]]], "]"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[LessEqual]", "n"]], "\[And]", RowBox[List[FractionBox[RowBox[List["n", "-", "m"]], "2"], "\[Element]", "Integers"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mi> k </mi> </msub> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mtext> </mtext> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mrow> <mi> m </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </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> </mrow> <mo> ⩵ </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <semantics> <mrow> <mfrac> <mrow> <mi> n </mi> <mo> - </mo> <mi> m </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mi> mod </mi> <mo> ⁢ </mo> <mn> 2 </mn> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <rem /> <apply> <times /> <apply> <plus /> <ci> $CellContext`n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> $CellContext`n1 </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </annotation-xml> </semantics> </msup> <mo> ⁢ </mo> <msup> <mn> 2 </mn> <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> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - 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</mo> <mi> m </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ∈ </mo> <semantics> <mi> ℕ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalN]", Function[Integers]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> FormBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <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> m </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> cos </ms> <ms> ( </ms> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> k </ms> </apply> <ms> ) </ms> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∏ </ms> <apply> <ci> RowBox </ci> <list> <ms> k </ms> <ms> = </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </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> </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> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> InterpretationBox </ci> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> - </ms> <ms> m </ms> </list> </apply> <ms> 2 </ms> </apply> <ms> </ms> <ms> mod </ms> <ms> </ms> <ms> 2 </ms> </list> </apply> <apply> <rem /> <apply> <times /> <apply> <plus /> <ci> $CellContext`n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> $CellContext`n1 </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> SuperscriptBox </ci> <ms> 2 </ms> <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> <apply> <ci> SuperscriptBox </ci> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <ms> - </ms> <ms> 1 </ms> </list> </apply> <ms> ) </ms> </list> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> ∑ </ms> <apply> <ci> RowBox </ci> <list> <ms> j </ms> <ms> = </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> </list> </apply> <ms> n </ms> </apply> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <ms> 1 </ms> <ms> 4 </ms> </apply> <apply> <ci> RowBox </ci> <list> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> 2 </ms> <apply> <ci> SubscriptBox </ci> <ms> k </ms> <ms> j </ms> </apply> </list> </apply> <ms> + </ms> <ms> 2 </ms> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </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> <apply> <ci> SubscriptBox </ci> <ms> z </ms> <ms> j </ms> </apply> </list> </apply> </list> </apply> <ms> ) </ms> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </list> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> <ms> /; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> ∈ </ms> <apply> <ci> TagBox </ci> <ms> ℕ </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> ∈ </ms> <apply> <ci> TagBox </ci> <ms> ℕ </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> ≤ </ms> <ms> n </ms> </list> </apply> <ms> ∧ </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> - </ms> <ms> m </ms> </list> </apply> <ms> 2 </ms> </apply> <ms> ∈ </ms> <apply> <ci> TagBox </ci> <ms> ℕ </ms> <apply> <ci> Function </ci> <integers /> </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