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 Cos

 http://functions.wolfram.com/01.07.21.2755.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18