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 Cos

 http://functions.wolfram.com/01.07.21.2456.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18