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 Cos

 http://functions.wolfram.com/01.07.21.2523.01

 Input Form

 Integrate[z^n Sin[d z + e]^m Cos[c z^2]^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[m, m/2] (-1 + Mod[m, 2]) Sum[Binomial[v, s] (((-I) c (2 s - v) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-I) c (2 s - v) z^2] + (I c (2 s - v) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, I c (2 s - v) z^2]), {s, 0, Floor[(1/2) (-1 + v)]}] + (2 Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[(-1)^k Binomial[m, k] (E^(I e (-2 k + m)) (I d (2 k - m))^(-1 - n) Gamma[1 + n, I d (2 k - m) z] + E^(I (2 e k - e m + m Pi)) (I d (-2 k + m))^(-1 - n) Gamma[1 + n, I d (-2 k + m) z]), {k, 0, Floor[(1/2) (-1 + m)]}])/I^m + ((-1)^n (I c)^(-1 - 2 n) Sum[(-1)^k Binomial[m, k] Sum[((2 s - v)^(-1 - 2 n) Binomial[v, s] ((-E^(4 I e (-k + m))) (I c (-2 s + v))^n Sum[(I d (k - m/2))^(n - q) (I (-2 d k + d m + 4 c s z - 2 c v z))^(1 + q) (-((I (-2 d k + d m + 4 c s z - 2 c v z)^2)/(c (2 s - v))))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (-2 d k + d m + 4 c s z - 2 c v z)^2)/(c (8 s - 4 v)))], {q, 0, n}] + E^(I (2 e m + m Pi + (d^2 (-2 k + m)^2)/ (4 c s - 2 c v))) (I c (2 s - v))^n Sum[2^(-n + q) (I d (-2 k + m))^(n - q) ((-I) (-2 d k + d m + 4 c s z - 2 c v z))^(1 + q) ((I (-2 d k + d m + 4 c s z - 2 c v z)^2)/(c (2 s - v)))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (-2 d k + d m + 4 c s z - 2 c v z)^2)/ (8 c s - 4 c v)], {q, 0, n}] - E^(I m (2 e + Pi)) (I c (-2 s + v))^n Sum[2^(-n + q) (I d (-2 k + m))^(n - q) (I (2 d k - d m + 4 c s z - 2 c v z))^(1 + q) (-((I (d (-2 k + m) + 2 c (-2 s + v) z)^2)/(c (2 s - v))))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (-2 d k + d m - 4 c s z + 2 c v z)^2)/(8 c s - 4 c v))], {q, 0, n}] + E^(I (4 e (-k + m) + (d^2 (-2 k + m)^2)/ (4 c s - 2 c v))) (I c (2 s - v))^n Sum[(I d (k - m/2))^ (n - q) (I (d (-2 k + m) + 2 c (-2 s + v) z))^(1 + q) ((I (d (-2 k + m) + 2 c (-2 s + v) z)^2)/(c (2 s - v)))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (-2 d k + d m - 4 c s z + 2 c v z)^2)/(8 c s - 4 c v)], {q, 0, n}]))/E^(I (e (-2 k + 3 m) + (d^2 (-2 k + m)^2)/ (c (8 s - 4 v)))), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}])/I^m) /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", FractionBox[RowBox[List["\[ImaginaryI]", " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "d", " ", "k"]], "+", RowBox[List["d", " ", "m"]], "-", RowBox[List["4", " ", "c", " ", "s", " ", "z"]], "+", RowBox[List["2", " ", "c", " ", "v", " ", "z"]]]], ")"]], "2"]]], RowBox[List[RowBox[List["8", " ", "c", " ", "s"]], "-", RowBox[List["4", " ", "c", " ", "v"]]]]]]], "]"]]]]]]]]]], ")"]]]]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "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 ( d z + e ) cos v ( c z 2 ) 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]]]]] + 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]]]]] ( 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 2 , - c ( 2 s - v ) z 2 ) ( - c ( 2 s - v ) z 2 ) 1 2 ( - n - 1 ) + ( c ( 2 s - v ) z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , c ( 2 s - v ) z 2 ) ) + 2 - 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]]]]] ( e ( m - 2 k ) Γ ( n + 1 , d ( 2 k - m ) z ) ( d ( 2 k - m ) ) - n - 1 + ( 2 e k - e m + m π ) ( d ( m - 2 k ) ) - n - 1 Γ ( n + 1 , d ( m - 2 k ) z ) ) + ( - 1 ) n - m ( c ) - 2 n - 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]]]]] s = 0 v - 1 2 - ( d 2 ( m - 2 k ) 2 c ( 8 s - 4 v ) + e ( 3 m - 2 k ) ) ( 2 s - v ) - 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]]]]] ( ( d 2 ( m - 2 k ) 2 4 c s - 2 c v + 4 e ( m - k ) ) ( q = 0 n ( d ( k - m 2 ) ) n - q ( ( d ( m - 2 k ) + 2 c ( v - 2 s ) z ) ) q + 1 ( ( d ( m - 2 k ) + 2 c ( v - 2 s ) z ) 2 c ( 2 s - 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 d k + d m - 4 c s z + 2 c v z ) 2 8 c s - 4 c v ) ) ( c ( 2 s - v ) ) n + ( d 2 ( m - 2 k ) 2 4 c s - 2 c v + 2 e m + m π ) ( q = 0 n 2 q - n ( d ( m - 2 k ) ) n - q ( - ( - 2 d k + d m + 4 c s z - 2 c v z ) ) q + 1 ( ( - 2 d k + d m + 4 c s z - 2 c v z ) 2 c ( 2 s - 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 d k + d m + 4 c s z - 2 c v z ) 2 8 c s - 4 c v ) ) ( c ( 2 s - v ) ) n - 4 e ( m - k ) ( c ( v - 2 s ) ) n q = 0 n ( d ( k - m 2 ) ) n - q ( ( - 2 d k + d m + 4 c s z - 2 c v z ) ) q + 1 ( - ( - 2 d k + d m + 4 c s z - 2 c v z ) 2 c ( 2 s - 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 d k + d m + 4 c s z - 2 c v z ) 2 c ( 8 s - 4 v ) ) - m ( 2 e + π ) ( c ( v - 2 s ) ) n q = 0 n 2 q - n ( d ( m - 2 k ) ) n - q ( ( 2 d k - d m + 4 c s z - 2 c v z ) ) q + 1 ( - ( d ( m - 2 k ) + 2 c ( v - 2 s ) z ) 2 c ( 2 s - 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 d k + d m - 4 c s z + 2 c v z ) 2 8 c s - 4 c v ) ) ) /; n m + v + Condition z z n d z e m c z 2 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 z n 1 Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 Binomial v s Gamma n 1 2 -1 -1 c 2 s -1 v z 2 -1 c 2 s -1 v z 2 1 2 -1 n -1 c 2 s -1 v z 2 1 2 -1 n -1 Gamma n 1 2 -1 c 2 s -1 v z 2 2 -1 m Binomial v v 2 -1 \$CellContext`v 2 -1 k 0 m -1 2 -1 -1 k Binomial m k e m -1 2 k Gamma n 1 d 2 k -1 m z d 2 k -1 m -1 n -1 2 e k -1 e m m d m -1 2 k -1 n -1 Gamma n 1 d m -1 2 k z -1 n -1 m c -2 n -1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 -1 d 2 m -1 2 k 2 c 8 s -1 4 v -1 e 3 m -1 2 k 2 s -1 v -2 n -1 Binomial v s d 2 m -1 2 k 2 4 c s -1 2 c v -1 4 e m -1 k q 0 n d k -1 m 2 -1 n -1 q d m -1 2 k 2 c v -1 2 s z q 1 d m -1 2 k 2 c v -1 2 s z 2 c 2 s -1 v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -2 d k d m -1 4 c s z 2 c v z 2 8 c s -1 4 c v -1 c 2 s -1 v n d 2 m -1 2 k 2 4 c s -1 2 c v -1 2 e m m q 0 n 2 q -1 n d m -1 2 k n -1 q -1 -2 d k d m 4 c s z -1 2 c v z q 1 -2 d k d m 4 c s z -1 2 c v z 2 c 2 s -1 v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -2 d k d m 4 c s z -1 2 c v z 2 8 c s -1 4 c v -1 c 2 s -1 v n -1 4 e m -1 k c v -1 2 s n q 0 n d k -1 m 2 -1 n -1 q -2 d k d m 4 c s z -1 2 c v z q 1 -1 -2 d k d m 4 c s z -1 2 c v z 2 c 2 s -1 v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 -2 d k d m 4 c s z -1 2 c v z 2 c 8 s -1 4 v -1 -1 m 2 e c v -1 2 s n q 0 n 2 q -1 n d m -1 2 k n -1 q 2 d k -1 d m 4 c s z -1 2 c v z q 1 -1 d m -1 2 k 2 c v -1 2 s z 2 c 2 s -1 v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 -2 d k d m -1 4 c s z 2 c v z 2 8 c s -1 4 c v -1 n m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18