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 Cos

 http://functions.wolfram.com/01.07.21.2505.01

 Input Form

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

 Standard Form

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"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 ) cos v ( c z ) 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]]]]] + - 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 , - b ( 2 k - m ) z 2 ) ( - b ( 2 k - m ) z 2 ) 1 2 ( - n - 1 ) + ( - b ( m - 2 k ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , - b ( m - 2 k ) 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]]]]] ( 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 , - c ( v - 2 s ) z ) ( - c ( v - 2 s ) ) - n - 1 + ( c ( v - 2 s ) ) - n - 1 Γ ( n + 1 , c ( v - 2 s ) z ) ) - 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 b - 1 4 ( c 2 ( v - 2 s ) 2 b ( 2 k - m ) + 2 m π ) ( m - 2 k ) ( b 2 ( m - 2 k ) 2 ) - n - 1 ( 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 k - 2 b m ( q = 0 n 2 q - n ( - c ( v - 2 s ) ) n - q ( ( c ( v - 2 s ) + 2 b ( m - 2 k ) z ) ) q + 1 ( ( c ( v - 2 s ) + 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 , ( - 2 c s + c v - 4 b k z + 2 b m z ) 2 8 b k - 4 b m ) ) ( b ( 2 k - m ) ) n - c 2 ( v - 2 s ) 2 4 b k - 2 b m ( q = 0 n 2 q - n ( c ( v - 2 s ) ) n - q ( - ( - 2 c s + c v + 4 b k z - 2 b m z ) ) q + 1 ( ( - 2 c s + c v + 4 b k z - 2 b 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 , ( - 2 c s + c v + 4 b k z - 2 b m z ) 2 8 b k - 4 b m ) ) ( b ( 2 k - m ) ) n + m π ( b ( m - 2 k ) ) n q = 0 n 2 q - n ( - c ( v - 2 s ) ) n - q ( ( - 2 c s + c v + 4 b k z - 2 b m z ) ) q + 1 ( - ( - 2 c s + c v + 4 b k z - 2 b 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 , - ( - 2 c s + c v + 4 b k z - 2 b m z ) 2 b ( 8 k - 4 m ) ) + m π ( b ( m - 2 k ) ) n q = 0 n 2 q - n ( c ( v - 2 s ) ) 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 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 , - ( - 2 c s + c v - 4 b k z + 2 b m z ) 2 8 b k - 4 b m ) ) ) /; n m + v + Condition z z n b z 2 m c z 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 -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 -1 b 2 k -1 m z 2 -1 b 2 k -1 m z 2 1 2 -1 n -1 -1 b m -1 2 k z 2 1 2 -1 n -1 Gamma n 1 2 -1 -1 b m -1 2 k z 2 2 Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 Binomial v s Gamma n 1 -1 c v -1 2 s z -1 c v -1 2 s -1 n -1 c v -1 2 s -1 n -1 Gamma n 1 c v -1 2 s z -1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 b -1 1 4 c 2 v -1 2 s 2 b 2 k -1 m -1 2 m m -1 2 k b 2 m -1 2 k 2 -1 n -1 Binomial v s -1 c 2 v -1 2 s 2 4 b k -1 2 b m -1 q 0 n 2 q -1 n -1 c v -1 2 s n -1 q c v -1 2 s 2 b m -1 2 k z q 1 c v -1 2 s 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 -2 c s c v -1 4 b k z 2 b m z 2 8 b k -1 4 b m -1 b 2 k -1 m n -1 c 2 v -1 2 s 2 4 b k -1 2 b m -1 q 0 n 2 q -1 n c v -1 2 s n -1 q -1 -2 c s c v 4 b k z -1 2 b m z q 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 Binomial n q Gamma q 1 2 -1 -2 c s c v 4 b k z -1 2 b m z 2 8 b k -1 4 b m -1 b 2 k -1 m n m b m -1 2 k n q 0 n 2 q -1 n -1 c v -1 2 s n -1 q -2 c s c v 4 b k z -1 2 b m z q 1 -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 Binomial n q Gamma q 1 2 -1 -1 -2 c s c v 4 b k z -1 2 b m z 2 b 8 k -1 4 m -1 m b m -1 2 k n q 0 n 2 q -1 n c v -1 2 s 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 k z 2 b 2 k -1 m -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 -2 c s c v -1 4 b k z 2 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