html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Sin

 http://functions.wolfram.com/01.06.20.0014.01

 Input Form

 D[Sin[z]^m, {z, n}] == Sum[(-1)^k Binomial[m, k] (m - 2 k)^n ((I^(n - m) + I^(m - n)) Sum[(-1)^j Binomial[m - 2 k, 2 j] Sin[z]^(2 j) Cos[z]^(m - 2 k - 2 j), {j, 0, Floor[m/2] - k}] + I (I^(n - m) - I^(m - n)) Sum[(-1)^j Binomial[m - 2 k, 2 j + 1] Sin[z]^(2 j + 1) Cos[z]^(m - 2 k - 2 j - 1), {j, 0, Floor[(m - 1)/2] - k}]), {k, 0, Floor[(m - 1)/2]}]/2^m /; Element[m, Integers] && m >= 0 && Element[n, Integers] && n > 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], "m"]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["m", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", "k"]]]], ")"]], "n"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["n", "-", "m"]]], "+", SuperscriptBox["\[ImaginaryI]", RowBox[List["m", "-", "n"]]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", FractionBox["m", "2"], "]"]], "-", "k"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["m", "-", RowBox[List["2", "k"]]]], ",", RowBox[List["2", " ", "j"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], RowBox[List["2", " ", "j"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], RowBox[List["m", "-", RowBox[List["2", "k"]], "-", RowBox[List["2", " ", "j"]]]]]]]]]]], "+", RowBox[List["\[ImaginaryI]", RowBox[List["(", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["n", "-", "m"]]], "-", SuperscriptBox["\[ImaginaryI]", RowBox[List["m", "-", "n"]]]]], ")"]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["m", "-", "1"]], "2"], "]"]], "-", "k"]]], " ", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["m", "-", RowBox[List["2", "k"]]]], ",", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], RowBox[List["m", "-", RowBox[List["2", "k"]], "-", RowBox[List["2", " ", "j"]], "-", "1"]]]]]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", ">", "0"]]]]]]]]

 MathML Form

 n sin m ( z ) z n 2 - m k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( m - 2 k ) n ( ( n - m - m - n ) j = 0 m - 1 2 - k ( - 1 ) j ( m - 2 k 2 j + 1 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] sin 2 j + 1 ( z ) cos - 2 j - 2 k + m - 1 ( z ) + ( n - m + m - n ) j = 0 m 2 - k ( - 1 ) j ( m - 2 k 2 j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox[RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], Identity, Rule[Editable, True]]], List[TagBox[RowBox[List["2", " ", "j"]], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] sin 2 j ( z ) cos - 2 j - 2 k + m ( z ) ) /; m n + Condition z n z m 2 -1 m k 0 m -1 2 -1 -1 k Binomial m k m -1 2 k n n -1 m -1 m -1 n j 0 m -1 2 -1 -1 k -1 j Binomial m -1 2 k 2 j 1 z 2 j 1 z -2 j -1 2 k m -1 n -1 m m -1 n j 0 m 2 -1 -1 k -1 j Binomial m -1 2 k 2 j z 2 j z -2 j -1 2 k m m n SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], SuperscriptBox[RowBox[List["Sin", "[", "z_", "]"]], "m_"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List["-", "m"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", FractionBox[RowBox[List["m", "-", "1"]], "2"], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ")"]], "n"], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["n", "-", "m"]]], "+", SuperscriptBox["\[ImaginaryI]", RowBox[List["m", "-", "n"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", FractionBox["m", "2"], "]"]], "-", "k"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ",", RowBox[List["2", " ", "j"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], RowBox[List["2", " ", "j"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], RowBox[List["m", "-", RowBox[List["2", " ", "k"]], "-", RowBox[List["2", " ", "j"]]]]]]]]]]], "+", RowBox[List["\[ImaginaryI]", " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ImaginaryI]", RowBox[List["n", "-", "m"]]], "-", SuperscriptBox["\[ImaginaryI]", RowBox[List["m", "-", "n"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["Floor", "[", FractionBox[RowBox[List["m", "-", "1"]], "2"], "]"]], "-", "k"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", RowBox[List["Binomial", "[", RowBox[List[RowBox[List["m", "-", RowBox[List["2", " ", "k"]]]], ",", RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]]], "]"]], " ", SuperscriptBox[RowBox[List["Sin", "[", "z", "]"]], RowBox[List[RowBox[List["2", " ", "j"]], "+", "1"]]], " ", SuperscriptBox[RowBox[List["Cos", "[", "z", "]"]], RowBox[List["m", "-", RowBox[List["2", " ", "k"]], "-", RowBox[List["2", " ", "j"]], "-", "1"]]]]]]]]]]], ")"]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", ">", "0"]]]]]]]]]]

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

 2007-05-02