html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cos

 http://functions.wolfram.com/01.07.21.2722.01

 Input Form

 Integrate[z^n E^(p Sqrt[z]) Sin[b z] Cos[c z]^v, z] == (I 2^(-2 - 2 n - v) (E^((I p^2)/(2 b)) Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p + 2 I b Sqrt[z])^(h + k) ((I (p + 2 I b Sqrt[z])^2)/b)^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p + 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (p + 2 I b Sqrt[z])^2)/(4 b)] + 2 I b Sqrt[(I (p + 2 I b Sqrt[z])^2)/b] Gamma[(1/2) (2 + h + k), (I (p + 2 I b Sqrt[z])^2)/(4 b)]), {k, 0, n}, {h, 0, k}] - Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p - 2 I b Sqrt[z])^(h + k) ((I (I p + 2 b Sqrt[z])^2)/b)^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p - 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (I p + 2 b Sqrt[z])^2)/(4 b)] - 2 I b Sqrt[(I (I p + 2 b Sqrt[z])^2)/b] Gamma[(1/2) (2 + h + k), (I (I p + 2 b Sqrt[z])^2)/(4 b)]), {k, 0, n}, {h, 0, k}] + (I b)^(2 n) b^2 E^((I p^2)/(4 b)) Sum[Binomial[v, s] (-((E^((I p^2)/(4 b + 8 c s - 4 c v)) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p + 2 I (b + 2 c s - c v) Sqrt[z])^(h + k) ((I (p + 2 I (b + 2 c s - c v) Sqrt[z])^2)/(b + 2 c s - c v))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p + 2 I (b + 2 c s - c v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (p + 2 I (b + 2 c s - c v) Sqrt[z])^2)/(4 (b + 2 c s - c v))] + 2 I (b + 2 c s - c v) Sqrt[ (I (p + 2 I (b + 2 c s - c v) Sqrt[z])^2)/(b + 2 c s - c v)] Gamma[(1/2) (2 + h + k), (I (p + 2 I (b + 2 c s - c v) Sqrt[z])^2)/(4 (b + 2 c s - c v))]), {k, 0, n}, {h, 0, k}])/(I (b + 2 c s - c v))^(2 n))/(b + 2 c s - c v)^2 + (E^((I p^2)/(-4 b - 8 c s + 4 c v)) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p - 2 I (b + 2 c s - c v) Sqrt[z])^(h + k) ((I (I p + 2 (b + 2 c s - c v) Sqrt[z])^2)/(b + 2 c s - c v))^( (1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p - 2 I (b + 2 c s - c v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (I p + 2 (b + 2 c s - c v) Sqrt[z])^2)/(4 (b + 2 c s - c v))] - 2 I (b + 2 c s - c v) Sqrt[ (I (I p + 2 (b + 2 c s - c v) Sqrt[z])^2)/(b + 2 c s - c v)] Gamma[(1/2) (2 + h + k), (I (I p + 2 (b + 2 c s - c v) Sqrt[z])^2)/(4 (b + 2 c s - c v))]), {k, 0, n}, {h, 0, k}])/((-I) (b + 2 c s - c v))^(2 n)/(b + 2 c s - c v)^2 + (((b - 2 c s + c v)^2)^(-1 - 2 n) ((-E^((I p^2)/(2 (b - 2 c s + c v)))) ((-I) (b - 2 c s + c v))^ (2 n) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p + 2 I (b - 2 c s + c v) Sqrt[z])^(h + k) ((I (p + 2 I (b - 2 c s + c v) Sqrt[z])^2)/(b - 2 c s + c v))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ( p (p + 2 I (b - 2 c s + c v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (p + 2 I (b - 2 c s + c v) Sqrt[z])^2)/ (4 (b - 2 c s + c v))] + 2 I (b - 2 c s + c v) Sqrt[(I (p + 2 I (b - 2 c s + c v) Sqrt[z])^2)/(b - 2 c s + c v)] Gamma[(1/2) (2 + h + k), (I (p + 2 I (b - 2 c s + c v) Sqrt[z])^2)/(4 (b - 2 c s + c v))]), {k, 0, n}, {h, 0, k}] + (I (b - 2 c s + c v))^(2 n) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p - 2 I (b - 2 c s + c v) Sqrt[z])^(h + k) ((I (I p + 2 (b - 2 c s + c v) Sqrt[z])^2)/(b - 2 c s + c v))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p - 2 I (b - 2 c s + c v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (I p + 2 (b - 2 c s + c v) Sqrt[z])^2)/(4 (b - 2 c s + c v))] - 2 I (b - 2 c s + c v) Sqrt[ (I (I p + 2 (b - 2 c s + c v) Sqrt[z])^2)/(b - 2 c s + c v)] Gamma[(1/2) (2 + h + k), (I (I p + 2 (b - 2 c s + c v) Sqrt[z])^2)/(4 (b - 2 c s + c v))]), {k, 0, n}, {h, 0, k}]))/E^((I p^2)/(4 (b - 2 c s + c v)))), {s, 0, Floor[(1/2) (-1 + v)]}]))/((I b)^(2 (1 + n)) E^((I p^2)/(4 b))) /; 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 ( b ) - 2 ( n + 1 ) - p 2 4 b ( b 2 p 2 4 b ( 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]]]]] ( - p 2 4 ( b - 2 c s + c v ) ( ( b - 2 c s + c v ) 2 ) - 2 n - 1 ( ( ( b - 2 c s + c v ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p - 2 ( b - 2 c s + c v ) z ) h + k ( ( p + 2 ( b - 2 c s + c v ) z ) 2 b - 2 c s + c v ) 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]]]]] ( p ( p - 2 ( b - 2 c s + c v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( p + 2 ( b - 2 c s + c v ) z ) 2 4 ( b - 2 c s + c v ) ) - 2 ( b - 2 c s + c v ) ( p + 2 ( b - 2 c s + c v ) z ) 2 b - 2 c s + c v Γ ( 1 2 ( h + k + 2 ) , ( p + 2 ( b - 2 c s + c v ) z ) 2 4 ( b - 2 c s + c v ) ) ) - p 2 2 ( b - 2 c s + c v ) ( - ( b - 2 c s + c v ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p + 2 ( b - 2 c s + c v ) z ) h + k ( ( p + 2 ( b - 2 c s + c v ) z ) 2 b - 2 c s + c v ) 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]]]]] ( p ( p + 2 ( b - 2 c s + c v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( p + 2 ( b - 2 c s + c v ) z ) 2 4 ( b - 2 c s + c v ) ) + 2 ( p + 2 ( b - 2 c s + c v ) z ) 2 b - 2 c s + c v ( b - 2 c s + c v ) Γ ( 1 2 ( h + k + 2 ) , ( p + 2 ( b - 2 c s + c v ) z ) 2 4 ( b - 2 c s + c v ) ) ) ) + 1 ( b + 2 c s - c v ) 2 ( p 2 - 4 b - 8 c s + 4 c v ( - ( b + 2 c s - c v ) ) - 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p - 2 ( b + 2 c s - c v ) z ) h + k ( ( p + 2 ( b + 2 c s - c v ) z ) 2 b + 2 c s - c v ) 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]]]]] ( p ( p - 2 ( b + 2 c s - c v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( p + 2 ( b + 2 c s - c v ) z ) 2 4 ( b + 2 c s - c v ) ) - 2 ( b + 2 c s - c v ) ( p + 2 ( b + 2 c s - c v ) z ) 2 b + 2 c s - c v Γ ( 1 2 ( h + k + 2 ) , ( p + 2 ( b + 2 c s - c v ) z ) 2 4 ( b + 2 c s - c v ) ) ) ) - 1 ( b + 2 c s - c v ) 2 ( p 2 4 b + 8 c s - 4 c v ( ( b + 2 c s - c v ) ) - 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p + 2 ( b + 2 c s - c v ) z ) h + k ( ( p + 2 ( b + 2 c s - c v ) z ) 2 b + 2 c s - c v ) 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]]]]] ( p ( p + 2 ( b + 2 c s - c v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( p + 2 ( b + 2 c s - c v ) z ) 2 4 ( b + 2 c s - c v ) ) + 2 ( p + 2 ( b + 2 c s - c v ) z ) 2 b + 2 c s - c v ( b + 2 c s - c v ) Γ ( 1 2 ( h + k + 2 ) , ( p + 2 ( b + 2 c s - c v ) z ) 2 4 ( b + 2 c s - c v ) ) ) ) ) ) ( b ) 2 n - ( 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 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p - 2 b z ) h + k ( ( 2 z b + p ) 2 b ) 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]]]]] ( p ( p - 2 b z ) Γ ( 1 2 ( h + k + 1 ) , ( 2 z b + p ) 2 4 b ) - 2 b ( 2 z b + p ) 2 b Γ ( 1 2 ( h + k + 2 ) , ( 2 z b + p ) 2 4 b ) ) + p 2 2 b ( 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 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p + 2 b z ) h + k ( ( p + 2 b z ) 2 b ) 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]]]]] ( p ( p + 2 b z ) Γ ( 1 2 ( h + k + 1 ) , ( p + 2 b z ) 2 4 b ) + 2 ( p + 2 b z ) 2 b b Γ ( 1 2 ( h + k + 2 ) , ( p + 2 b z ) 2 4 b ) ) ) /; v + n Condition z z n p z 1 2 b z c z v 2 -2 n -1 v -2 b -2 n 1 -1 p 2 4 b -1 b 2 p 2 4 b -1 s 0 v -1 2 -1 Binomial v s -1 p 2 4 b -1 2 c s c v -1 b -1 2 c s c v 2 -2 n -1 b -1 2 c s c v 2 n h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p -1 2 b -1 2 c s c v z 1 2 h k p 2 b -1 2 c s c v z 1 2 2 b -1 2 c s c v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p -1 2 b -1 2 c s c v z 1 2 Gamma 1 2 h k 1 p 2 b -1 2 c s c v z 1 2 2 4 b -1 2 c s c v -1 -1 2 b -1 2 c s c v p 2 b -1 2 c s c v z 1 2 2 b -1 2 c s c v -1 1 2 Gamma 1 2 h k 2 p 2 b -1 2 c s c v z 1 2 2 4 b -1 2 c s c v -1 -1 p 2 2 b -1 2 c s c v -1 -1 b -1 2 c s c v 2 n h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p 2 b -1 2 c s c v z 1 2 h k p 2 b -1 2 c s c v z 1 2 2 b -1 2 c s c v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p 2 b -1 2 c s c v z 1 2 Gamma 1 2 h k 1 p 2 b -1 2 c s c v z 1 2 2 4 b -1 2 c s c v -1 2 p 2 b -1 2 c s c v z 1 2 2 b -1 2 c s c v -1 1 2 b -1 2 c s c v Gamma 1 2 h k 2 p 2 b -1 2 c s c v z 1 2 2 4 b -1 2 c s c v -1 1 b 2 c s -1 c v 2 -1 p 2 -4 b -1 8 c s 4 c v -1 -1 b 2 c s -1 c v -2 n h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p -1 2 b 2 c s -1 c v z 1 2 h k p 2 b 2 c s -1 c v z 1 2 2 b 2 c s -1 c v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p -1 2 b 2 c s -1 c v z 1 2 Gamma 1 2 h k 1 p 2 b 2 c s -1 c v z 1 2 2 4 b 2 c s -1 c v -1 -1 2 b 2 c s -1 c v p 2 b 2 c s -1 c v z 1 2 2 b 2 c s -1 c v -1 1 2 Gamma 1 2 h k 2 p 2 b 2 c s -1 c v z 1 2 2 4 b 2 c s -1 c v -1 -1 1 b 2 c s -1 c v 2 -1 p 2 4 b 8 c s -1 4 c v -1 b 2 c s -1 c v -2 n h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p 2 b 2 c s -1 c v z 1 2 h k p 2 b 2 c s -1 c v z 1 2 2 b 2 c s -1 c v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p 2 b 2 c s -1 c v z 1 2 Gamma 1 2 h k 1 p 2 b 2 c s -1 c v z 1 2 2 4 b 2 c s -1 c v -1 2 p 2 b 2 c s -1 c v z 1 2 2 b 2 c s -1 c v -1 1 2 b 2 c s -1 c v Gamma 1 2 h k 2 p 2 b 2 c s -1 c v z 1 2 2 4 b 2 c s -1 c v -1 b 2 n -1 Binomial v v 2 -1 \$CellContext`v 2 -1 h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p -1 2 b z 1 2 h k 2 z 1 2 b p 2 b -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p -1 2 b z 1 2 Gamma 1 2 h k 1 2 z 1 2 b p 2 4 b -1 -1 2 b 2 z 1 2 b p 2 b -1 1 2 Gamma 1 2 h k 2 2 z 1 2 b p 2 4 b -1 p 2 2 b -1 Binomial v v 2 -1 \$CellContext`v 2 -1 h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p 2 b z 1 2 h k p 2 b z 1 2 2 b -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p 2 b z 1 2 Gamma 1 2 h k 1 p 2 b z 1 2 2 4 b -1 2 p 2 b z 1 2 2 b -1 1 2 b Gamma 1 2 h k 2 p 2 b z 1 2 2 4 b -1 v SuperPlus n [/itex]

 Rule Form

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

 2002-12-18