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/01.07.21.2446.01

 Input Form

 Integrate[z^n Sin[b z] Cos[c Sqrt[z]]^v, z] == 2^(-2 - 2 n - v) b^(-2 - 2 n) ((-2^(1 + 2 n)) b^(1 + n) Binomial[v, v/2] (I^n Gamma[1 + n, (-I) b z] + (-I)^n Gamma[1 + n, I b z]) (1 - Mod[v, 2]) + I (-1)^n Sum[(Binomial[v, s] (-Sum[(-1)^(-h + k) 4^k (I c (2 s - v))^(-h - k + 2 n) (I (2 c s - c v + 2 b Sqrt[z]))^(h + k) (-((I (2 c s - c v + 2 b Sqrt[z])^2)/b))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (c (2 s - v) (2 c s - c v + 2 b Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (2 c s - c v + 2 b Sqrt[z])^2)/(4 b))] - 2 I b Sqrt[-((I (2 c s - c v + 2 b Sqrt[z])^2)/b)] Gamma[(1/2) (2 + h + k), -((I (2 c s - c v + 2 b Sqrt[z])^2)/(4 b))]), {k, 0, n}, {h, 0, k}] + E^((I c^2 (-2 s + v)^2)/(2 b)) Sum[(-1)^(-h + k) 4^k (I c (-2 s + v))^(-h - k + 2 n) (I (-2 c s + c v - 2 b Sqrt[z]))^(h + k) ((I (2 c s - c v + 2 b Sqrt[z])^2)/b)^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (c (2 s - v) (2 c s - c v + 2 b Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (2 c s - c v + 2 b Sqrt[z])^2)/(4 b)] + 2 I b Sqrt[(I (2 c s - c v + 2 b Sqrt[z])^2)/b] Gamma[(1/2) (2 + h + k), (I (2 c s - c v + 2 b Sqrt[z])^2)/(4 b)]), {k, 0, n}, {h, 0, k}] - Sum[(-1)^(-h + k) 4^k (I c (-2 s + v))^(-h - k + 2 n) ((-I) (2 c s - c v - 2 b Sqrt[z]))^(h + k) (-((I (-2 c s + c v + 2 b Sqrt[z])^2)/b))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (c (2 s - v) (2 c s - c v - 2 b Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (-2 c s + c v + 2 b Sqrt[z])^2)/(4 b))] - 2 I b Sqrt[-((I (-2 c s + c v + 2 b Sqrt[z])^2)/b)] Gamma[(1/2) (2 + h + k), -((I (-2 c s + c v + 2 b Sqrt[z])^2)/ (4 b))]), {k, 0, n}, {h, 0, k}] + E^((I c^2 (-2 s + v)^2)/(2 b)) Sum[(-1)^(-h + k) 4^k (I c (2 s - v))^(-h - k + 2 n) (I (2 c s - c v - 2 b Sqrt[z]))^ (h + k) ((I (-2 c s + c v + 2 b Sqrt[z])^2)/b)^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (c (2 s - v) (2 c s - c v - 2 b Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (-2 c s + c v + 2 b Sqrt[z])^2)/(4 b)] + 2 I b Sqrt[(I (-2 c s + c v + 2 b Sqrt[z])^2)/b] Gamma[(1/2) (2 + h + k), (I (-2 c s + c v + 2 b Sqrt[z])^2)/ (4 b)]), {k, 0, n}, {h, 0, k}]))/ E^((I c^2 (-2 s + v)^2)/(4 b)), {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 ) cos v ( c z ) z 2 - 2 n - v - 2 b - 2 n - 2 ( ( - 1 ) 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]]]]] ( c 2 ( v - 2 s ) 2 2 b k = 0 n h = 0 k ( - 1 ) k - h 4 k ( c ( 2 s - v ) ) - h - k + 2 n ( ( - 2 z b + 2 c s - c v ) ) h + k ( ( 2 z b - 2 c s + c v ) 2 b ) 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]]]]] ( c ( 2 s - v ) ( - 2 z b + 2 c s - c v ) Γ ( 1 2 ( h + k + 1 ) , ( 2 z b - 2 c s + c v ) 2 4 b ) + 2 ( 2 z b - 2 c s + c v ) 2 b b Γ ( 1 2 ( h + k + 2 ) , ( 2 z b - 2 c s + c v ) 2 4 b ) ) + c 2 ( v - 2 s ) 2 2 b k = 0 n h = 0 k ( - 1 ) k - h 4 k ( c ( v - 2 s ) ) - h - k + 2 n ( ( - 2 z b - 2 c s + c v ) ) h + k ( ( 2 z b + 2 c s - c v ) 2 b ) 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]]]]] ( c ( 2 s - v ) ( 2 z b + 2 c s - c v ) Γ ( 1 2 ( h + k + 1 ) , ( 2 z b + 2 c s - c v ) 2 4 b ) + 2 ( 2 z b + 2 c s - c v ) 2 b b Γ ( 1 2 ( h + k + 2 ) , ( 2 z b + 2 c s - c v ) 2 4 b ) ) - k = 0 n h = 0 k ( - 1 ) k - h 4 k ( c ( v - 2 s ) ) - h - k + 2 n ( - ( - 2 z b + 2 c s - c v ) ) h + k ( - ( 2 z b - 2 c s + c v ) 2 b ) 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]]]]] ( c ( 2 s - v ) ( - 2 z b + 2 c s - c v ) Γ ( 1 2 ( h + k + 1 ) , - ( 2 z b - 2 c s + c v ) 2 4 b ) - 2 b - ( 2 z b - 2 c s + c v ) 2 b Γ ( 1 2 ( h + k + 2 ) , - ( 2 z b - 2 c s + c v ) 2 4 b ) ) - k = 0 n h = 0 k ( - 1 ) k - h 4 k ( c ( 2 s - v ) ) - h - k + 2 n ( ( 2 z b + 2 c s - c v ) ) h + k ( - ( 2 z b + 2 c s - c v ) 2 b ) 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]]]]] ( c ( 2 s - v ) ( 2 z b + 2 c s - c v ) Γ ( 1 2 ( h + k + 1 ) , - ( 2 z b + 2 c s - c v ) 2 4 b ) - 2 b - ( 2 z b + 2 c s - c v ) 2 b Γ ( 1 2 ( h + k + 2 ) , - ( 2 z b + 2 c s - c v ) 2 4 b ) ) ) - 2 2 n + 1 b 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]]]]] ( Γ ( n + 1 , b z ) ( - ) n + n Γ ( n + 1 , - b z ) ) ( 1 - v mod 2 \$CellContext`v 2 ) ) /; n v + Condition z z n b z c z 1 2 v 2 -2 n -1 v -2 b -2 n -2 -1 n s 0 v -1 2 -1 -1 c 2 v -1 2 s 2 4 b -1 Binomial v s c 2 v -1 2 s 2 2 b -1 h 0 k k 0 n -1 k -1 h 4 k c 2 s -1 v -1 h -1 k 2 n -2 z 1 2 b 2 c s -1 c v h k 2 z 1 2 b -1 2 c s c v 2 b -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c 2 s -1 v -2 z 1 2 b 2 c s -1 c v Gamma 1 2 h k 1 2 z 1 2 b -1 2 c s c v 2 4 b -1 2 2 z 1 2 b -1 2 c s c v 2 b -1 1 2 b Gamma 1 2 h k 2 2 z 1 2 b -1 2 c s c v 2 4 b -1 c 2 v -1 2 s 2 2 b -1 h 0 k k 0 n -1 k -1 h 4 k c v -1 2 s -1 h -1 k 2 n -2 z 1 2 b -1 2 c s c v h k 2 z 1 2 b 2 c s -1 c v 2 b -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c 2 s -1 v 2 z 1 2 b 2 c s -1 c v Gamma 1 2 h k 1 2 z 1 2 b 2 c s -1 c v 2 4 b -1 2 2 z 1 2 b 2 c s -1 c v 2 b -1 1 2 b Gamma 1 2 h k 2 2 z 1 2 b 2 c s -1 c v 2 4 b -1 -1 h 0 k k 0 n -1 k -1 h 4 k c v -1 2 s -1 h -1 k 2 n -1 -2 z 1 2 b 2 c s -1 c v h k -1 2 z 1 2 b -1 2 c s c v 2 b -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c 2 s -1 v -2 z 1 2 b 2 c s -1 c v Gamma 1 2 h k 1 -1 2 z 1 2 b -1 2 c s c v 2 4 b -1 -1 2 b -1 2 z 1 2 b -1 2 c s c v 2 b -1 1 2 Gamma 1 2 h k 2 -1 2 z 1 2 b -1 2 c s c v 2 4 b -1 -1 h 0 k k 0 n -1 k -1 h 4 k c 2 s -1 v -1 h -1 k 2 n 2 z 1 2 b 2 c s -1 c v h k -1 2 z 1 2 b 2 c s -1 c v 2 b -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c 2 s -1 v 2 z 1 2 b 2 c s -1 c v Gamma 1 2 h k 1 -1 2 z 1 2 b 2 c s -1 c v 2 4 b -1 -1 2 b -1 2 z 1 2 b 2 c s -1 c v 2 b -1 1 2 Gamma 1 2 h k 2 -1 2 z 1 2 b 2 c s -1 c v 2 4 b -1 -1 2 2 n 1 b n 1 Binomial v v 2 -1 Gamma n 1 b z -1 n n Gamma n 1 -1 b z 1 -1 \$CellContext`v 2 n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18