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/01.07.21.2467.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18