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; }

 Cos

 http://functions.wolfram.com/01.07.21.2507.01

 Input Form

 Integrate[z^n Sin[b z^2 + e]^m Cos[c z]^v, z] == 2^(-1 - m - v) ((1/(1 + n)) (2 z^(1 + n) Binomial[m, m/2] Binomial[v, v/2] (-1 + Mod[m, 2]) (-1 + Mod[v, 2])) - z^(1 + n) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[((-1)^k (b^2 (-2 k + m)^2 z^4)^((1/2) (-1 - n)) Binomial[m, k] (E^(I (4 e k + m Pi)) (I b (2 k - m) z^2)^((1 + n)/2) Gamma[(1 + n)/2, (-I) b (2 k - m) z^2] + E^(2 I e m) ((-I) b (2 k - m) z^2)^((1 + n)/2) Gamma[(1 + n)/2, I b (2 k - m) z^2]))/E^((1/2) I (2 e (2 k + m) + m Pi)), {k, 0, Floor[(1/2) (-1 + m)]}] - 2 Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, s] (((-I) c (2 s - v))^(-1 - n) Gamma[1 + n, (-I) c (2 s - v) z] + (I c (2 s - v))^(-1 - n) Gamma[1 + n, I c (2 s - v) z]), {s, 0, Floor[(1/2) (-1 + v)]}] - Sum[(-1)^k Binomial[m, k] Sum[(Binomial[v, s] (E^(I (e (4 k - 2 m) + m Pi - (-2 c s + c v)^2/ (8 b k - 4 b m))) (I (2 b k - b m))^(-1 - n) Sum[2^(-n + q) ((-I) (-2 c s + c v))^(n - q) (I (c (-2 s + v) + 2 (2 b k - b m) z))^(1 + q) (-((I (c (-2 s + v) + 2 (2 b k - b m) z)^2)/(2 b k - b m)))^( (1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (c (-2 s + v) + 2 (2 b k - b m) z)^2)/(8 b k - 4 b m))], {q, 0, n}] + E^((I (-2 c s + c v)^2)/(8 b k - 4 b m)) ((-I) (2 b k - b m))^(-1 - n) Sum[2^(-n + q) (I (-2 c s + c v))^( n - q) ((-I) (c (-2 s + v) + 2 (2 b k - b m) z))^(1 + q) ((I (c (-2 s + v) + 2 (2 b k - b m) z)^2)/(2 b k - b m))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (c (-2 s + v) + 2 (2 b k - b m) z)^2)/(8 b k - 4 b m)], {q, 0, n}] + E^(I (e (4 k - 2 m) + m Pi - (c^2 (-2 s + v)^2)/ (8 b k - 4 b m))) (I (2 b k - b m))^(-1 - n) Sum[2^(-n + q) ((-I) (2 c s - c v))^(n - q) (I (2 c s - c v + 4 b k z - 2 b m z))^(1 + q) (-((I (c (-2 s + v) + 2 (-2 b k + b m) z)^2)/(2 b k - b m)))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (c (-2 s + v) + 2 (-2 b k + b m) z)^2)/(8 b k - 4 b m))], {q, 0, n}] + E^((I c^2 (-2 s + v)^2)/(8 b k - 4 b m)) ((-I) (2 b k - b m))^ (-1 - n) Sum[2^(-n + q) (I (2 c s - c v))^(n - q) ((-I) (2 c s - c v + 4 b k z - 2 b m z))^(1 + q) ((I (c (-2 s + v) + 2 (-2 b k + b m) z)^2)/(2 b k - b m))^( (1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (c (-2 s + v) + 2 (-2 b k + b m) z)^2)/(8 b k - 4 b m)], {q, 0, n}]))/E^(I (2 e k - e m)), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}]/ I^m) /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["c", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "s"]], "+", "v"]], ")"]]]], "+", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]], ")"]], " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]]]]]]], "]"]]]]]]]]]], ")"]]]]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 z n sin m ( b z 2 + e ) cos v ( c z ) z 2 - m - v - 1 ( 2 z n + 1 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) n + 1 - z n + 1 ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) k = 0 m - 1 2 ( - 1 ) k - 1 2 ( π m + 2 e ( 2 k + m ) ) ( b 2 ( m - 2 k ) 2 z 4 ) 1 2 ( - n - 1 ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 e m Γ ( n + 1 2 , b ( 2 k - m ) z 2 ) ( - b ( 2 k - m ) z 2 ) n + 1 2 + ( 4 e k + m π ) ( b ( 2 k - m ) z 2 ) n + 1 2 Γ ( n + 1 2 , - b ( 2 k - m ) z 2 ) ) - 2 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( Γ ( n + 1 , - c ( 2 s - v ) z ) ( - c ( 2 s - v ) ) - n - 1 + ( c ( 2 s - v ) ) - n - 1 Γ ( n + 1 , c ( 2 s - v ) z ) ) - - m k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] s = 0 v - 1 2 - ( 2 e k - e m ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( c 2 ( v - 2 s ) 2 8 b k - 4 b m ( q = 0 n 2 q - n ( ( 2 c s - c v ) ) n - q ( - ( 2 c s - c v + 4 b k z - 2 b m z ) ) q + 1 ( ( c ( v - 2 s ) + 2 ( b m - 2 b k ) z ) 2 2 b k - b m ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( c ( v - 2 s ) + 2 ( b m - 2 b k ) z ) 2 8 b k - 4 b m ) ) ( - ( 2 b k - b m ) ) - n - 1 + ( c v - 2 c s ) 2 8 b k - 4 b m ( q = 0 n 2 q - n ( ( c v - 2 c s ) ) n - q ( - ( c ( v - 2 s ) + 2 ( 2 b k - b m ) z ) ) q + 1 ( ( c ( v - 2 s ) + 2 ( 2 b k - b m ) z ) 2 2 b k - b m ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( c ( v - 2 s ) + 2 ( 2 b k - b m ) z ) 2 8 b k - 4 b m ) ) ( - ( 2 b k - b m ) ) - n - 1 + ( - c 2 ( v - 2 s ) 2 8 b k - 4 b m + e ( 4 k - 2 m ) + m π ) ( ( 2 b k - b m ) ) - n - 1 q = 0 n 2 q - n ( - ( 2 c s - c v ) ) n - q ( ( 2 c s - c v + 4 b k z - 2 b m z ) ) q + 1 ( - ( c ( v - 2 s ) + 2 ( b m - 2 b k ) z ) 2 2 b k - b m ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( c ( v - 2 s ) + 2 ( b m - 2 b k ) z ) 2 8 b k - 4 b m ) + ( - ( c v - 2 c s ) 2 8 b k - 4 b m + e ( 4 k - 2 m ) + m π ) ( ( 2 b k - b m ) ) - n - 1 q = 0 n 2 q - n ( - ( c v - 2 c s ) ) n - q ( ( c ( v - 2 s ) + 2 ( 2 b k - b m ) z ) ) q + 1 ( - ( c ( v - 2 s ) + 2 ( 2 b k - b m ) z ) 2 2 b k - b m ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( c ( v - 2 s ) + 2 ( 2 b k - b m ) z ) 2 8 b k - 4 b m ) ) ) /; n m + v + Condition z z n b z 2 e m c z v 2 -1 m -1 v -1 2 z n 1 Binomial m m 2 -1 Binomial v v 2 -1 \$CellContext`m 2 -1 \$CellContext`v 2 -1 n 1 -1 -1 z n 1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k -1 1 2 m 2 e 2 k m b 2 m -1 2 k 2 z 4 1 2 -1 n -1 Binomial m k 2 e m Gamma n 1 2 -1 b 2 k -1 m z 2 -1 b 2 k -1 m z 2 n 1 2 -1 4 e k m b 2 k -1 m z 2 n 1 2 -1 Gamma n 1 2 -1 -1 b 2 k -1 m z 2 -1 2 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 Binomial v s Gamma n 1 -1 c 2 s -1 v z -1 c 2 s -1 v -1 n -1 c 2 s -1 v -1 n -1 Gamma n 1 c 2 s -1 v z -1 -1 m k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 -1 2 e k -1 e m Binomial v s c 2 v -1 2 s 2 8 b k -1 4 b m -1 q 0 n 2 q -1 n 2 c s -1 c v n -1 q -1 2 c s -1 c v 4 b k z -1 2 b m z q 1 c v -1 2 s 2 b m -1 2 b k z 2 2 b k -1 b m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 c v -1 2 s 2 b m -1 2 b k z 2 8 b k -1 4 b m -1 -1 2 b k -1 b m -1 n -1 c v -1 2 c s 2 8 b k -1 4 b m -1 q 0 n 2 q -1 n c v -1 2 c s n -1 q -1 c v -1 2 s 2 2 b k -1 b m z q 1 c v -1 2 s 2 2 b k -1 b m z 2 2 b k -1 b m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 c v -1 2 s 2 2 b k -1 b m z 2 8 b k -1 4 b m -1 -1 2 b k -1 b m -1 n -1 -1 c 2 v -1 2 s 2 8 b k -1 4 b m -1 e 4 k -1 2 m m 2 b k -1 b m -1 n -1 q 0 n 2 q -1 n -1 2 c s -1 c v n -1 q 2 c s -1 c v 4 b k z -1 2 b m z q 1 -1 c v -1 2 s 2 b m -1 2 b k z 2 2 b k -1 b m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 c v -1 2 s 2 b m -1 2 b k z 2 8 b k -1 4 b m -1 -1 c v -1 2 c s 2 8 b k -1 4 b m -1 e 4 k -1 2 m m 2 b k -1 b m -1 n -1 q 0 n 2 q -1 n -1 c v -1 2 c s n -1 q c v -1 2 s 2 2 b k -1 b m z q 1 -1 c v -1 2 s 2 2 b k -1 b m z 2 2 b k -1 b m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 c v -1 2 s 2 2 b k -1 b m z 2 8 b k -1 4 b m -1 n m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18