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/01.07.21.2549.01

 Input Form

 Integrate[z^n Sin[d z]^m Cos[c z^2 + f z]^v, z] == 2^(-1 - m - v) ((1/(1 + n)) (2 z^(1 + n) Binomial[m, m/2] Binomial[v, v/2] (-1 + Mod[m, 2]) (-1 + Mod[v, 2])) + 2 I^(1 - m) d^(-1 - 2 n) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k (2 k - m)^(-1 - 2 n) Binomial[m, k] ((-1)^(m - 1) (I d (2 k - m))^n Gamma[1 + n, (-I) d (2 k - m) z] + ((-I) d (2 k - m))^n Gamma[1 + n, I d (2 k - m) z]), {k, 0, Floor[(1/2) (-1 + m)]}] + (-1)^n (I c)^(-1 - n) Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[(Binomial[v, s] (Sum[(-1)^q 2^(-n + q) (I f)^(n - q) (I (f + 2 c z))^(1 + q) (-((I (2 s - v) (f + 2 c z)^2)/c))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (2 s - v) (f + 2 c z)^2)/(4 c))], {q, 0, n}] + E^((I f^2 (2 s - v))/(2 c)) Sum[(-1)^q 2^(-n + q) (I f)^(n - q) (I (f + 2 c z))^(1 + q) ((I (2 s - v) (f + 2 c z)^2)/c)^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (2 s - v) (f + 2 c z)^2)/(4 c)], {q, 0, n}]))/ E^((I f^2 (2 s - v))/(4 c)), {s, 0, Floor[(1/2) (-1 + v)]}] - Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (E^(I (m Pi - (2 d k - d m - 2 f s + f v)^2/(-8 c s + 4 c v))) (I (-2 c s + c v))^(-1 - n) Sum[2^(-n + q) ((-I) (2 d k - d m - 2 f s + f v))^(n - q) (I (d (2 k - m) + f (-2 s + v) + 2 (-2 c s + c v) z))^(1 + q) (-((1/(-2 c s + c v)) (I (d (2 k - m) + f (-2 s + v) + 2 (-2 c s + c v) z)^2)))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (d (2 k - m) + f (-2 s + v) + 2 (-2 c s + c v) z)^2)/(-8 c s + 4 c v))], {q, 0, n}] + E^((I (2 d k - d m - 2 f s + f v)^2)/(-8 c s + 4 c v)) ((-I) (-2 c s + c v))^(-1 - n) Sum[2^(-n + q) (I (2 d k - d m - 2 f s + f v))^(n - q) ((-I) (d (2 k - m) + f (-2 s + v) + 2 (-2 c s + c v) z))^(1 + q) ((I (d (2 k - m) + f (-2 s + v) + 2 (-2 c s + c v) z)^2)/( -2 c s + c v))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (d (2 k - m) + f (-2 s + v) + 2 (-2 c s + c v) z)^2)/(-8 c s + 4 c v)], {q, 0, n}] + E^(I (m Pi - (d (-2 k + m) + f (-2 s + v))^2/(8 c s - 4 c v))) (I (2 c s - c v))^(-1 - n) Sum[2^(-n + q) ((-I) (2 d k - d m + 2 f s - f v))^(n - q) (I (2 d k - d m + 2 f s - f v + 4 c s z - 2 c v z))^(1 + q) (-((1/(2 c s - c v)) (I (d (-2 k + m) + f (-2 s + v) + 2 (-2 c s + c v) z)^2)))^ ((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I (d (-2 k + m) + f (-2 s + v) + 2 (-2 c s + c v) z)^2)/ (8 c s - 4 c v))], {q, 0, n}] + E^((I (d (-2 k + m) + f (-2 s + v))^2)/(8 c s - 4 c v)) ((-I) (2 c s - c v))^(-1 - n) Sum[2^(-n + q) (I (2 d k - d m + 2 f s - f v))^(n - q) ((-I) (2 d k - d m + 2 f s - f v + 4 c s z - 2 c v z))^(1 + q) ((1/(2 c s - c v)) (I (d (-2 k + m) + f (-2 s + v) + 2 (-2 c s + c v) z)^2))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, (I (d (-2 k + m) + f (-2 s + v) + 2 (-2 c s + c v) z)^2)/(8 c s - 4 c v)], {q, 0, n}]), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}]/ I^m) /; Element[n, Integers] && n >= 0 && Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 z n sin m ( d z ) cos v ( c z 2 + f z ) z 2 - m - v - 1 ( 2 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]]]]] ( 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]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) n + 1 + 2 1 - m d - 2 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]]]]] ( 1 - v mod 2 \$CellContext`v 2 ) k = 0 m - 1 2 ( - 1 ) k ( 2 k - m ) - 2 n - 1 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( Γ ( n + 1 , d ( 2 k - m ) z ) ( - d ( 2 k - m ) ) n + ( - 1 ) m - 1 ( d ( 2 k - m ) ) n Γ ( n + 1 , - d ( 2 k - m ) z ) ) + ( - 1 ) n ( c ) - 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 - f 2 ( 2 s - v ) 4 c ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( f 2 ( 2 s - v ) 2 c q = 0 n ( - 1 ) q 2 q - n ( f ) n - q ( ( f + 2 c z ) ) q + 1 ( ( 2 s - v ) ( f + 2 c z ) 2 c ) 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 s - v ) ( f + 2 c z ) 2 4 c ) + q = 0 n ( - 1 ) q 2 q - n ( f ) n - q ( ( f + 2 c z ) ) q + 1 ( - ( 2 s - v ) ( f + 2 c z ) 2 c ) 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 s - v ) ( f + 2 c z ) 2 4 c ) ) - - m 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]]]]] 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]]]]] ( ( 2 d k - d m - 2 f s + f v ) 2 4 c v - 8 c s ( q = 0 n 2 q - n ( ( 2 d k - d m - 2 f s + f v ) ) n - q ( - ( d ( 2 k - m ) + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) ) q + 1 ( ( d ( 2 k - m ) + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) 2 c v - 2 c s ) 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 , ( d ( 2 k - m ) + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) 2 4 c v - 8 c s ) ) ( - ( c v - 2 c s ) ) - n - 1 + ( m π - ( 2 d k - d m - 2 f s + f v ) 2 4 c v - 8 c s ) ( ( c v - 2 c s ) ) - n - 1 q = 0 n 2 q - n ( - ( 2 d k - d m - 2 f s + f v ) ) n - q ( ( d ( 2 k - m ) + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) ) q + 1 ( - ( d ( 2 k - m ) + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) 2 c v - 2 c s ) 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 , - ( d ( 2 k - m ) + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) 2 4 c v - 8 c s ) + ( d ( m - 2 k ) + f ( v - 2 s ) ) 2 8 c s - 4 c v ( - ( 2 c s - c v ) ) - n - 1 q = 0 n 2 q - n ( ( 2 d k - d m + 2 f s - f v ) ) n - q ( - ( 2 d k - d m + 2 f s - f v + 4 c s z - 2 c v z ) ) q + 1 ( ( d ( m - 2 k ) + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) 2 2 c s - c v ) 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 , ( d ( m - 2 k ) + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) 2 8 c s - 4 c v ) + ( m π - ( d ( m - 2 k ) + f ( v - 2 s ) ) 2 8 c s - 4 c v ) ( ( 2 c s - c v ) ) - n - 1 q = 0 n 2 q - n ( - ( 2 d k - d m + 2 f s - f v ) ) n - q ( ( 2 d k - d m + 2 f s - f v + 4 c s z - 2 c v z ) ) q + 1 ( - ( d ( m - 2 k ) + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) 2 2 c s - c v ) 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 , - ( d ( m - 2 k ) + f ( v - 2 s ) + 2 ( c v - 2 c s ) z ) 2 8 c s - 4 c v ) ) ) /; n m + v + Condition z z n d z m c z 2 f z v 2 -1 m -1 v -1 2 z n 1 Binomial m m 2 -1 Binomial v v 2 -1 \$CellContext`m 2 -1 \$CellContext`v 2 -1 n 1 -1 2 1 -1 m d -2 n -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k 2 k -1 m -2 n -1 Binomial m k Gamma n 1 d 2 k -1 m z -1 d 2 k -1 m n -1 m -1 d 2 k -1 m n Gamma n 1 -1 d 2 k -1 m z -1 n c -1 n -1 Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 -1 f 2 2 s -1 v 4 c -1 Binomial v s f 2 2 s -1 v 2 c -1 q 0 n -1 q 2 q -1 n f n -1 q f 2 c z q 1 2 s -1 v f 2 c z 2 c -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 2 s -1 v f 2 c z 2 4 c -1 q 0 n -1 q 2 q -1 n f n -1 q f 2 c z q 1 -1 2 s -1 v f 2 c z 2 c -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 2 s -1 v f 2 c z 2 4 c -1 -1 -1 m k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s 2 d k -1 d m -1 2 f s f v 2 4 c v -1 8 c s -1 q 0 n 2 q -1 n 2 d k -1 d m -1 2 f s f v n -1 q -1 d 2 k -1 m f v -1 2 s 2 c v -1 2 c s z q 1 d 2 k -1 m f v -1 2 s 2 c v -1 2 c s z 2 c v -1 2 c s -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 d 2 k -1 m f v -1 2 s 2 c v -1 2 c s z 2 4 c v -1 8 c s -1 -1 c v -1 2 c s -1 n -1 m -1 2 d k -1 d m -1 2 f s f v 2 4 c v -1 8 c s -1 c v -1 2 c s -1 n -1 q 0 n 2 q -1 n -1 2 d k -1 d m -1 2 f s f v n -1 q d 2 k -1 m f v -1 2 s 2 c v -1 2 c s z q 1 -1 d 2 k -1 m f v -1 2 s 2 c v -1 2 c s z 2 c v -1 2 c s -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 d 2 k -1 m f v -1 2 s 2 c v -1 2 c s z 2 4 c v -1 8 c s -1 d m -1 2 k f v -1 2 s 2 8 c s -1 4 c v -1 -1 2 c s -1 c v -1 n -1 q 0 n 2 q -1 n 2 d k -1 d m 2 f s -1 f v n -1 q -1 2 d k -1 d m 2 f s -1 f v 4 c s z -1 2 c v z q 1 d m -1 2 k f v -1 2 s 2 c v -1 2 c s z 2 2 c s -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 d m -1 2 k f v -1 2 s 2 c v -1 2 c s z 2 8 c s -1 4 c v -1 m -1 d m -1 2 k f v -1 2 s 2 8 c s -1 4 c v -1 2 c s -1 c v -1 n -1 q 0 n 2 q -1 n -1 2 d k -1 d m 2 f s -1 f v n -1 q 2 d k -1 d m 2 f s -1 f v 4 c s z -1 2 c v z q 1 -1 d m -1 2 k f v -1 2 s 2 c v -1 2 c s z 2 2 c s -1 c v -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 d m -1 2 k f v -1 2 s 2 c v -1 2 c s z 2 8 c s -1 4 c v -1 n m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18