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 Cos

 http://functions.wolfram.com/01.07.21.2753.01

 Input Form

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

 Standard Form

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" ", "s"]], "+", RowBox[List["c", " ", "v"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", "z"]]]], ")"]], "2"], "p"], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]], "-", RowBox[List["2", " ", "c", " ", "s"]], "+", RowBox[List["c", " ", "v"]], "+", RowBox[List["2", " ", "\[ImaginaryI]", " ", "p", " ", "z"]]]], ")"]], "2"], RowBox[List["4", " ", "p"]]]]], "]"]]]]]]]]]], ")"]]]]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18