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Cos






Mathematica Notation

Traditional Notation









Elementary Functions > Cos[z] > Transformations > Products, sums, and powers of the direct function > Products involving the direct function





http://functions.wolfram.com/01.07.16.0114.01









  


  










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 + 1)/2, 2] + 1)/2^n * Sum[(-1)^Sum[(2 Subscript[k, j] + 2)/4, {j, m + 1, n}]* Sin[Sum[Subscript[k, j] Subscript[z, j], {j, 1, n}]], {Subscript[k, 1], -1, 1, 2}, {Subscript[k, 2], -1, 1, 2}, …, {Subscript[k, n], -1, 1, 2}]/; n ∈ Integers && m ∈ Integers && n >= 0 && m >= 0 && m <= n && (n - m + 1)/2 ∈ 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[RowBox[List["Mod", "[", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "m", "-", "1"]], ")"]], "/", "2"]], ",", "2"]], "]"]], "+", "1"]]], 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["Sin", "[", 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", "+", "1"]], "2"], "\[Element]", "Integers"]]]]]]]]










MathML Form







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</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> &#916; </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> &#8721; </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> &#916; </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> &#8230; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </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> &#916; </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> &#8721; </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> sin </ms> <ms> ( </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> UnderoverscriptBox </ci> <ms> &#8721; </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> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8469; </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8469; </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <ms> m </ms> <ms> &#8804; </ms> <ms> n </ms> </list> </apply> <ms> &#8743; </ms> <apply> <ci> RowBox </ci> <list> <apply> <ci> FractionBox </ci> <apply> <ci> RowBox </ci> <list> <ms> n </ms> <ms> - </ms> <ms> m </ms> <ms> + </ms> <ms> 1 </ms> </list> </apply> <ms> 2 </ms> </apply> <ms> &#8712; </ms> <apply> <ci> TagBox </ci> <ms> &#8469; </ms> <apply> <ci> Function </ci> <integers /> </apply> </apply> </list> </apply> </list> </apply> </list> </apply> <ci> TraditionalForm </ci> </apply> </annotation-xml> </semantics> </math>










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2002-12-18





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