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/01.07.21.2759.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18