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/01.07.21.2728.01

 Input Form

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

 Rule Form

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

 2002-12-18