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 Cos

 http://functions.wolfram.com/01.07.21.2751.01

 Input Form

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

 Rule Form

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

 2002-12-18