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/01.07.21.2724.01

 Input Form

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