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/01.07.21.2492.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18