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/01.07.21.2531.01

 Input Form

 Integrate[z^n Sin[b z^2 + d z]^m Cos[c z^2]^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[m, m/2] (-1 + Mod[m, 2]) Sum[Binomial[v, s] (((-I) c (2 s - v) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-I) c (2 s - v) z^2] + (I c (2 s - v) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, I c (2 s - v) z^2]), {s, 0, Floor[(1/2) (-1 + v)]}] + (Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[((-1)^k (I b (2 k - m))^(-1 - n) Binomial[m, k] (E^((1/2) I ((d^2 (-2 k + m))/b + 2 m Pi)) Sum[2^(-n + q) (I d)^(n - q) (-2 k + m)^(1 + n) ((-I) (d + 2 b z))^ (1 + q) (-((I (2 k - m) (d + 2 b z)^2)/b))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (2 k - m) (d + 2 b z)^2)/ (4 b))], {q, 0, n}] + Sum[2^(-n + q) (I d)^(n - q) (-2 k + m)^(1 + n) ((-I) (d + 2 b z))^(1 + q) ((I (2 k - m) (d + 2 b z)^2)/b)^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (2 k - m) (d + 2 b z)^2)/(4 b)], {q, 0, n}]))/ E^((I d^2 (-2 k + m))/(4 b)), {k, 0, Floor[(1/2) (-1 + m)]}])/I^m - Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] ((I E^(I (m Pi - (d^2 (-2 k + m)^2)/(8 b k - 4 b m - 8 c s + 4 c v))) Sum[2^(-n + q) ((-I) d (2 k - m))^(n - q) (-((I (d (-2 k + m) + 2 (-2 b k + b m + 2 c s - c v) z)^2)/ (2 b k - b m - 2 c s + c v)))^((1/2) (-1 - q)) (I (d (2 k - m) + 2 (2 b k - b m - 2 c s + c v) z))^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, -((I (d (-2 k + m) + 2 (-2 b k + b m + 2 c s - c v) z)^2)/(8 b k - 4 b m - 8 c s + 4 c v))], {q, 0, n}])/ (I (2 b k - b m - 2 c s + c v))^n/(-2 b k + b m + 2 c s - c v) + E^((I d^2 (-2 k + m)^2)/(8 b k - 4 b m - 8 c s + 4 c v)) ((-I) (2 b k - b m - 2 c s + c v))^(-1 - n) Sum[2^(-n + q) (I d (2 k - m))^(n - q) ((I (d (-2 k + m) + 2 (-2 b k + b m + 2 c s - c v) z)^2)/( 2 b k - b m - 2 c s + c v))^((1/2) (-1 - q)) ((-I) (d (2 k - m) + 2 (2 b k - b m - 2 c s + c v) z))^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (I (d (-2 k + m) + 2 (-2 b k + b m + 2 c s - c v) z)^2)/(8 b k - 4 b m - 8 c s + 4 c v)], {q, 0, n}] + (I E^(I (m Pi - (d^2 (-2 k + m)^2)/(8 b k - 4 b m + 8 c s - 4 c v))) Sum[2^(-n + q) ((-I) d (2 k - m))^(n - q) (I (2 d k - d m + 4 b k z - 2 b m z + 4 c s z - 2 c v z))^ (1 + q) (-((I (d (-2 k + m) + 2 (b (-2 k + m) + c (-2 s + v)) z)^2)/(2 b k - b m + 2 c s - c v)))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (d (-2 k + m) + 2 (b (-2 k + m) + c (-2 s + v)) z)^2)/(8 b k - 4 b m + 8 c s - 4 c v))], {q, 0, n}])/ (I (2 b k - b m + 2 c s - c v))^n/(b (-2 k + m) + c (-2 s + v)) + E^((I d^2 (-2 k + m)^2)/(8 b k - 4 b m + 8 c s - 4 c v)) ((-I) (2 b k - b m + 2 c s - c v))^(-1 - n) Sum[(I d (k - m/2))^(n - q) ((-I) (2 d k - d m + 4 b k z - 2 b m z + 4 c s z - 2 c v z))^(1 + q) ((I (d (-2 k + m) + 2 (b (-2 k + m) + c (-2 s + v)) z)^2)/( 2 b k - b m + 2 c s - c v))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (d (-2 k + m) + 2 (b (-2 k + m) + c (-2 s + v)) z)^2)/(8 b k - 4 b m + 8 c s - 4 c v)], {q, 0, n}]), {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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 MathML Form

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

 Rule Form

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

 2002-12-18