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 Cos

 http://functions.wolfram.com/01.07.21.2726.01

 Input Form

 Integrate[z^n E^(p z) Sin[b z] Cos[c Sqrt[z]]^v, z] == 2^(-1 - v) I Binomial[v, v/2] (((-I) b - p)^(-1 - n) Gamma[1 + n, ((-I) b - p) z] - (I b - p)^(-1 - n) Gamma[1 + n, (I b - p) z]) (1 - Mod[v, 2]) + 2^(-1 - v - 2 n - 1) I Sum[Binomial[v, s] ((E^((c^2 (-2 s + v)^2)/(4 ((-I) b + p))) Sum[(-1)^(-h + i) 4^i ((-I) c (-2 s + v))^(-h - i + 2 n) ((-I) c (-2 s + v) + 2 ((-I) b + p) Sqrt[z])^(h + i) (-(((-I) c (-2 s + v) + 2 ((-I) b + p) Sqrt[z])^2/((-I) b + p)))^ ((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((-I) c (-2 s + v) ((-I) c (-2 s + v) + 2 ((-I) b + p) Sqrt[z]) Gamma[(1/2) (1 + h + i), -(((-I) c (-2 s + v) + 2 ((-I) b + p) Sqrt[z])^2/(4 ((-I) b + p)))] + 2 ((-I) b + p) Sqrt[-(((-I) c (-2 s + v) + 2 ((-I) b + p) Sqrt[z])^2/ ((-I) b + p))] Gamma[(1/2) (2 + h + i), -(((-I) c (-2 s + v) + 2 ((-I) b + p) Sqrt[z])^2/ (4 ((-I) b + p)))]), {i, 0, n}, {h, 0, i}])/ ((-I) b + p)^(2 (1 + n)) + (E^((c^2 (-2 s + v)^2)/(4 ((-I) b + p))) Sum[(-1)^(-h + i) 4^i (I c (-2 s + v))^(-h - i + 2 n) (I c (-2 s + v) + 2 ((-I) b + p) Sqrt[z])^(h + i) (-((I c (-2 s + v) + 2 ((-I) b + p) Sqrt[z])^2/((-I) b + p)))^ ((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] (I c (-2 s + v) (I c (-2 s + v) + 2 ((-I) b + p) Sqrt[z]) Gamma[(1/2) (1 + h + i), -((I c (-2 s + v) + 2 ((-I) b + p) Sqrt[z])^2/(4 ((-I) b + p)))] + 2 ((-I) b + p) Sqrt[-((I c (-2 s + v) + 2 ((-I) b + p) Sqrt[z])^2/((-I) b + p))] Gamma[(1/2) (2 + h + i), -((I c (-2 s + v) + 2 ((-I) b + p) Sqrt[z])^2/(4 ((-I) b + p)))]), {i, 0, n}, {h, 0, i}])/((-I) b + p)^(2 (1 + n)) - (E^((c^2 (-2 s + v)^2)/(4 (I b + p))) Sum[(-1)^(-h + i) 4^i ((-I) c (-2 s + v))^(-h - i + 2 n) ((-I) c (-2 s + v) + 2 (I b + p) Sqrt[z])^(h + i) (-(((-I) c (-2 s + v) + 2 (I b + p) Sqrt[z])^2/(I b + p)))^ ((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] ((-I) c (-2 s + v) ((-I) c (-2 s + v) + 2 (I b + p) Sqrt[z]) Gamma[(1/2) (1 + h + i), -(((-I) c (-2 s + v) + 2 (I b + p) Sqrt[z])^2/(4 (I b + p)))] + 2 (I b + p) Sqrt[-(((-I) c (-2 s + v) + 2 (I b + p) Sqrt[z])^2/(I b + p))] Gamma[(1/2) (2 + h + i), -(((-I) c (-2 s + v) + 2 (I b + p) Sqrt[z])^2/(4 (I b + p)))]), {i, 0, n}, {h, 0, i}])/ (I b + p)^(2 (1 + n)) - (E^((c^2 (-2 s + v)^2)/(4 (I b + p))) Sum[(-1)^(-h + i) 4^i (I c (-2 s + v))^(-h - i + 2 n) (I c (-2 s + v) + 2 (I b + p) Sqrt[z])^(h + i) (-((I c (-2 s + v) + 2 (I b + p) Sqrt[z])^2/(I b + p)))^ ((1/2) (-1 - h - i)) Binomial[i, h] Binomial[n, i] (I c (-2 s + v) (I c (-2 s + v) + 2 (I b + p) Sqrt[z]) Gamma[(1/2) (1 + h + i), -((I c (-2 s + v) + 2 (I b + p) Sqrt[z])^2/(4 (I b + p)))] + 2 (I b + p) Sqrt[-((I c (-2 s + v) + 2 (I b + p) Sqrt[z])^2/(I b + p))] Gamma[(1/2) (2 + h + i), -((I c (-2 s + v) + 2 (I b + p) Sqrt[z])^2/(4 (I b + p)))]), {i, 0, n}, {h, 0, i}])/ (I b + p)^(2 (1 + n))), {s, 0, Floor[(1/2) (-1 + v)]}] /; 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 - v - 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]]]]] ( ( - b - p ) - n - 1 Γ ( n + 1 , ( - b - p ) z ) - ( b - p ) - n - 1 Γ ( n + 1 , ( b - p ) z ) ) ( 1 - v mod 2 \$CellContext`v 2 ) + 2 - 2 n - v - 1 - 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]]]]] ( c 2 ( v - 2 s ) 2 4 ( - b + p ) ( i = 0 n h = 0 i ( - 1 ) i - h 4 i ( - c ( v - 2 s ) ) - h - i + 2 n ( 2 ( - b + p ) z - c ( v - 2 s ) ) h + i ( - ( 2 ( - b + p ) z - c ( v - 2 s ) ) 2 - b + 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 ( - b + p ) - ( 2 ( - b + p ) z - c ( v - 2 s ) ) 2 - b + p Γ ( 1 2 ( h + i + 2 ) , - ( 2 ( - b + p ) z - c ( v - 2 s ) ) 2 4 ( - b + p ) ) - c ( v - 2 s ) ( 2 ( - b + p ) z - c ( v - 2 s ) ) Γ ( 1 2 ( h + i + 1 ) , - ( 2 ( - b + p ) z - c ( v - 2 s ) ) 2 4 ( - b + p ) ) ) ) ( - b + p ) - 2 ( n + 1 ) + c 2 ( v - 2 s ) 2 4 ( - b + p ) ( i = 0 n h = 0 i ( - 1 ) i - h 4 i ( c ( v - 2 s ) ) - h - i + 2 n ( 2 z ( - b + p ) + c ( v - 2 s ) ) h + i ( - ( 2 z ( - b + p ) + c ( v - 2 s ) ) 2 - b + 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]]]]] ( c ( v - 2 s ) ( 2 z ( - b + p ) + c ( v - 2 s ) ) Γ ( 1 2 ( h + i + 1 ) , - ( 2 z ( - b + p ) + c ( v - 2 s ) ) 2 4 ( - b + p ) ) + 2 - ( 2 z ( - b + p ) + c ( v - 2 s ) ) 2 - b + p ( - b + p ) Γ ( 1 2 ( h + i + 2 ) , - ( 2 z ( - b + p ) + c ( v - 2 s ) ) 2 4 ( - b + p ) ) ) ) ( - b + p ) - 2 ( n + 1 ) - c 2 ( v - 2 s ) 2 4 ( b + p ) ( b + p ) - 2 ( n + 1 ) i = 0 n h = 0 i ( - 1 ) i - h 4 i ( - c ( v - 2 s ) ) - h - i + 2 n ( 2 ( b + p ) z - c ( v - 2 s ) ) h + i ( - ( 2 ( b + p ) z - c ( v - 2 s ) ) 2 b + 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 ( b + p ) - ( 2 ( b + p ) z - c ( v - 2 s ) ) 2 b + p Γ ( 1 2 ( h + i + 2 ) , - ( 2 ( b + p ) z - c ( v - 2 s ) ) 2 4 ( b + p ) ) - c ( v - 2 s ) ( 2 ( b + p ) z - c ( v - 2 s ) ) Γ ( 1 2 ( h + i + 1 ) , - ( 2 ( b + p ) z - c ( v - 2 s ) ) 2 4 ( b + p ) ) ) - c 2 ( v - 2 s ) 2 4 ( b + p ) ( b + p ) - 2 ( n + 1 ) i = 0 n h = 0 i ( - 1 ) i - h 4 i ( c ( v - 2 s ) ) - h - i + 2 n ( 2 z ( b + p ) + c ( v - 2 s ) ) h + i ( - ( 2 z ( b + p ) + c ( v - 2 s ) ) 2 b + 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]]]]] ( c ( v - 2 s ) ( 2 z ( b + p ) + c ( v - 2 s ) ) Γ ( 1 2 ( h + i + 1 ) , - ( 2 z ( b + p ) + c ( v - 2 s ) ) 2 4 ( b + p ) ) + 2 - ( 2 z ( b + p ) + c ( v - 2 s ) ) 2 b + p ( b + p ) Γ ( 1 2 ( h + i + 2 ) , - ( 2 z ( b + p ) + c ( v - 2 s ) ) 2 4 ( b + p ) ) ) ) /; v + n Condition z z n p z b z c z 1 2 v 2 -1 v -1 Binomial v v 2 -1 -1 b -1 p -1 n -1 Gamma n 1 -1 b -1 p z -1 b -1 p -1 n -1 Gamma n 1 b -1 p z 1 -1 \$CellContext`v 2 2 -2 n -1 v -1 -1 s 0 v -1 2 -1 Binomial v s c 2 v -1 2 s 2 4 -1 b p -1 h 0 i i 0 n -1 i -1 h 4 i -1 c v -1 2 s -1 h -1 i 2 n 2 -1 b p z 1 2 -1 c v -1 2 s h i -1 2 -1 b p z 1 2 -1 c v -1 2 s 2 -1 b p -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i 2 -1 b p -1 2 -1 b p z 1 2 -1 c v -1 2 s 2 -1 b p -1 1 2 Gamma 1 2 h i 2 -1 2 -1 b p z 1 2 -1 c v -1 2 s 2 4 -1 b p -1 -1 c v -1 2 s 2 -1 b p z 1 2 -1 c v -1 2 s Gamma 1 2 h i 1 -1 2 -1 b p z 1 2 -1 c v -1 2 s 2 4 -1 b p -1 -1 b p -2 n 1 c 2 v -1 2 s 2 4 -1 b p -1 h 0 i i 0 n -1 i -1 h 4 i c v -1 2 s -1 h -1 i 2 n 2 z 1 2 -1 b p c v -1 2 s h i -1 2 z 1 2 -1 b p c v -1 2 s 2 -1 b p -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i c v -1 2 s 2 z 1 2 -1 b p c v -1 2 s Gamma 1 2 h i 1 -1 2 z 1 2 -1 b p c v -1 2 s 2 4 -1 b p -1 2 -1 2 z 1 2 -1 b p c v -1 2 s 2 -1 b p -1 1 2 -1 b p Gamma 1 2 h i 2 -1 2 z 1 2 -1 b p c v -1 2 s 2 4 -1 b p -1 -1 b p -2 n 1 -1 c 2 v -1 2 s 2 4 b p -1 b p -2 n 1 h 0 i i 0 n -1 i -1 h 4 i -1 c v -1 2 s -1 h -1 i 2 n 2 b p z 1 2 -1 c v -1 2 s h i -1 2 b p z 1 2 -1 c v -1 2 s 2 b p -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i 2 b p -1 2 b p z 1 2 -1 c v -1 2 s 2 b p -1 1 2 Gamma 1 2 h i 2 -1 2 b p z 1 2 -1 c v -1 2 s 2 4 b p -1 -1 c v -1 2 s 2 b p z 1 2 -1 c v -1 2 s Gamma 1 2 h i 1 -1 2 b p z 1 2 -1 c v -1 2 s 2 4 b p -1 -1 c 2 v -1 2 s 2 4 b p -1 b p -2 n 1 h 0 i i 0 n -1 i -1 h 4 i c v -1 2 s -1 h -1 i 2 n 2 z 1 2 b p c v -1 2 s h i -1 2 z 1 2 b p c v -1 2 s 2 b p -1 1 2 -1 h -1 i -1 Binomial i h Binomial n i c v -1 2 s 2 z 1 2 b p c v -1 2 s Gamma 1 2 h i 1 -1 2 z 1 2 b p c v -1 2 s 2 4 b p -1 2 -1 2 z 1 2 b p c v -1 2 s 2 b p -1 1 2 b p Gamma 1 2 h i 2 -1 2 z 1 2 b p c v -1 2 s 2 4 b p -1 v SuperPlus n [/itex]

 Rule Form

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

 2002-12-18