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 Cos

 http://functions.wolfram.com/01.07.21.2730.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18