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/01.07.21.2429.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18