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/01.07.21.2439.01

 Input Form

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

 Standard Form

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 MathML Form

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