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/01.07.21.2719.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18