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/01.07.21.2524.01

 Input Form

 Integrate[z^n Sin[d z + e]^m Cos[c Sqrt[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 Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[(-1)^u Binomial[m, u] (E^(I e (m - 2 u)) ((-I) d (m - 2 u))^(-1 - n) Gamma[1 + n, (-I) d (m - 2 u) z] + ((-1)^m (I d (m - 2 u))^(-1 - n) Gamma[1 + n, I d (m - 2 u) z])/E^(I e (m - 2 u))), {u, 0, Floor[(1/2) (-1 + m)]}])/I^m + 4 Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[Binomial[v, s] (Gamma[2 (1 + n), (-I) c (2 s - v) Sqrt[z]]/((-I) c (2 s - v))^ (2 (1 + n)) + Gamma[2 (1 + n), I c (2 s - v) Sqrt[z]]/ (I c (2 s - v))^(2 (1 + n))), {s, 0, Floor[(1/2) (-1 + v)]}] + Sum[(-1)^u Binomial[m, u] Sum[(-E^((-I) e (m - 2 u) - (I c^2 (-2 s + v)^2)/(4 d (m - 2 u)))) (d^2 (m - 2 u)^2)^(-1 - 2 n) Binomial[v, s] (E^(2 I e (m - 2 u)) ((-I) d (m - 2 u))^(2 n) Sum[I (-1)^(-h + k) 4^k (I c (2 s - v))^(-h - k + 2 n) (I (c (2 s - v) + 2 d (m - 2 u) Sqrt[z]))^(h + k) (-((I (c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2)/(d (m - 2 u))))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (I c (2 s - v) (c (2 s - v) + 2 d (m - 2 u) Sqrt[z]) Gamma[ (1/2) (1 + h + k), -((I (c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^ 2)/(4 d (m - 2 u)))] + 2 d (m - 2 u) Sqrt[ -((I (c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2)/(d (m - 2 u)))] Gamma[(1/2) (2 + h + k), -((I (c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2)/(4 d (m - 2 u)))]), {k, 0, n}, {h, 0, k}] + (-1)^m E^((I c^2 (-2 s + v)^2)/(2 d (m - 2 u))) (I d (m - 2 u))^(2 n) Sum[I (-1)^(-h + k) 4^k (I c (-2 s + v))^ (-h - k + 2 n) (I (c (-2 s + v) - 2 d (m - 2 u) Sqrt[z]))^ (h + k) ((I (c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2)/(d (m - 2 u)))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (I c (-2 s + v) (c (-2 s + v) - 2 d (m - 2 u) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2)/(4 d (m - 2 u))] - 2 d (m - 2 u) Sqrt[(I (c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2)/ (d (m - 2 u))] Gamma[(1/2) (2 + h + k), (I (c (2 s - v) + 2 d (m - 2 u) Sqrt[z])^2)/(4 d (m - 2 u))]), {k, 0, n}, {h, 0, k}] + E^(2 I e (m - 2 u)) ((-I) d (m - 2 u))^(2 n) Sum[(-1)^(-h + k) 4^k (I c (-2 s + v))^ (-h - k + 2 n) (I (c (-2 s + v) + 2 d (m - 2 u) Sqrt[z]))^ (h + k) (-((I (c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2)/ (d (m - 2 u))))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((-c) (-2 s + v) (c (-2 s + v) + 2 d (m - 2 u) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2)/ (4 d (m - 2 u)))] + 2 I d (m - 2 u) Sqrt[ -((I (c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2)/ (d (m - 2 u)))] Gamma[(1/2) (2 + h + k), -((I (c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2)/ (4 d (m - 2 u)))]), {k, 0, n}, {h, 0, k}] + (-1)^m E^((I c^2 (-2 s + v)^2)/(2 d (m - 2 u))) (I d (m - 2 u))^(2 n) Sum[I (-1)^(-h + k) 4^k (I c (2 s - v))^ (-h - k + 2 n) (I (c (2 s - v) - 2 d (m - 2 u) Sqrt[z]))^ (h + k) ((I (c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2)/(d (m - 2 u)))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (I c (2 s - v) (c (2 s - v) - 2 d (m - 2 u) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2)/(4 d (m - 2 u))] - 2 d (m - 2 u) Sqrt[(I (c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2)/ (d (m - 2 u))] Gamma[(1/2) (2 + h + k), (I (c (-2 s + v) + 2 d (m - 2 u) Sqrt[z])^2)/ (4 d (m - 2 u))]), {k, 0, n}, {h, 0, k}]), {s, 0, Floor[(1/2) (-1 + v)]}], {u, 0, Floor[(1/2) (-1 + m)]}]/ (I^m 4^n)) /; 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 + e ) cos v ( c z ) z 2 - m - v - 1 ( 2 z n + 1 ( m mod 2 \$CellContext`m 2 - 1 ) ( v mod 2 \$CellContext`v 2 - 1 ) 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]]]]] + 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 ) u = 0 m - 1 2 ( - 1 ) u ( m u ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["u", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( e ( m - 2 u ) Γ ( n + 1 , - d ( m - 2 u ) z ) ( - d ( m - 2 u ) ) - n - 1 + ( - 1 ) m - e ( m - 2 u ) ( d ( m - 2 u ) ) - n - 1 Γ ( n + 1 , d ( m - 2 u ) z ) ) + 4 ( 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 ( 2 ( n + 1 ) , - c ( 2 s - v ) z ) ( - c ( 2 s - v ) ) - 2 ( n + 1 ) + ( c ( 2 s - v ) ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , c ( 2 s - v ) z ) ) + - m 4 - n u = 0 m - 1 2 ( - 1 ) u ( m u ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["u", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] s = 0 v - 1 2 - - c 2 ( v - 2 s ) 2 4 d ( m - 2 u ) - e ( m - 2 u ) ( d 2 ( m - 2 u ) 2 ) - 2 n - 1 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 e ( m - 2 u ) ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( c ( 2 s - v ) ) - h - k + 2 n ( ( 2 d z ( m - 2 u ) + c ( 2 s - v ) ) ) h + k ( - ( 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 d ( m - 2 u ) ) 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 d z ( m - 2 u ) + c ( 2 s - v ) ) Γ ( 1 2 ( h + k + 1 ) , - ( 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 4 d ( m - 2 u ) ) + 2 d ( m - 2 u ) - ( 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 d ( m - 2 u ) Γ ( 1 2 ( h + k + 2 ) , - ( 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 4 d ( m - 2 u ) ) ) ) ( - d ( m - 2 u ) ) 2 n + 2 e ( m - 2 u ) ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( c ( v - 2 s ) ) - h - k + 2 n ( ( 2 d z ( m - 2 u ) + c ( v - 2 s ) ) ) h + k ( - ( 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 d ( m - 2 u ) ) 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 ( m - 2 u ) - ( 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 d ( m - 2 u ) Γ ( 1 2 ( h + k + 2 ) , - ( 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 4 d ( m - 2 u ) ) - c ( v - 2 s ) ( 2 d z ( m - 2 u ) + c ( v - 2 s ) ) Γ ( 1 2 ( h + k + 1 ) , - ( 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 4 d ( m - 2 u ) ) ) ) ( - d ( m - 2 u ) ) 2 n + ( - 1 ) m c 2 ( v - 2 s ) 2 2 d ( m - 2 u ) ( d ( m - 2 u ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( c ( v - 2 s ) ) - h - k + 2 n ( ( c ( v - 2 s ) - 2 d ( m - 2 u ) z ) ) h + k ( ( 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 d ( m - 2 u ) ) 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 ( v - 2 s ) ( c ( v - 2 s ) - 2 d ( m - 2 u ) z ) Γ ( 1 2 ( h + k + 1 ) , ( 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 4 d ( m - 2 u ) ) - 2 d ( m - 2 u ) ( 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 d ( m - 2 u ) Γ ( 1 2 ( h + k + 2 ) , ( 2 d z ( m - 2 u ) + c ( 2 s - v ) ) 2 4 d ( m - 2 u ) ) ) + ( - 1 ) m c 2 ( v - 2 s ) 2 2 d ( m - 2 u ) ( d ( m - 2 u ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( c ( 2 s - v ) ) - h - k + 2 n ( ( c ( 2 s - v ) - 2 d ( m - 2 u ) z ) ) h + k ( ( 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 d ( m - 2 u ) ) 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 ) ( c ( 2 s - v ) - 2 d ( m - 2 u ) z ) Γ ( 1 2 ( h + k + 1 ) , ( 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 4 d ( m - 2 u ) ) - 2 d ( m - 2 u ) ( 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 d ( m - 2 u ) Γ ( 1 2 ( h + k + 2 ) , ( 2 d z ( m - 2 u ) + c ( v - 2 s ) ) 2 4 d ( m - 2 u ) ) ) ) ) /; n m + v + Condition z z n d z e m c z 1 2 v 2 -1 m -1 v -1 2 z n 1 \$CellContext`m 2 -1 \$CellContext`v 2 -1 n 1 -1 Binomial m m 2 -1 Binomial v v 2 -1 2 -1 m Binomial v v 2 -1 \$CellContext`v 2 -1 u 0 m -1 2 -1 -1 u Binomial m u e m -1 2 u Gamma n 1 -1 d m -1 2 u z -1 d m -1 2 u -1 n -1 -1 m -1 e m -1 2 u d m -1 2 u -1 n -1 Gamma n 1 d m -1 2 u z 4 Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 Binomial v s Gamma 2 n 1 -1 c 2 s -1 v z 1 2 -1 c 2 s -1 v -2 n 1 c 2 s -1 v -2 n 1 Gamma 2 n 1 c 2 s -1 v z 1 2 -1 m 4 -1 n u 0 m -1 2 -1 -1 u Binomial m u s 0 v -1 2 -1 -1 -1 c 2 v -1 2 s 2 4 d m -1 2 u -1 -1 e m -1 2 u d 2 m -1 2 u 2 -2 n -1 Binomial v s 2 e m -1 2 u h 0 k k 0 n -1 k -1 h 4 k c 2 s -1 v -1 h -1 k 2 n 2 d z 1 2 m -1 2 u c 2 s -1 v h k -1 2 d z 1 2 m -1 2 u c 2 s -1 v 2 d m -1 2 u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c 2 s -1 v 2 d z 1 2 m -1 2 u c 2 s -1 v Gamma 1 2 h k 1 -1 2 d z 1 2 m -1 2 u c 2 s -1 v 2 4 d m -1 2 u -1 2 d m -1 2 u -1 2 d z 1 2 m -1 2 u c 2 s -1 v 2 d m -1 2 u -1 1 2 Gamma 1 2 h k 2 -1 2 d z 1 2 m -1 2 u c 2 s -1 v 2 4 d m -1 2 u -1 -1 d m -1 2 u 2 n 2 e m -1 2 u h 0 k k 0 n -1 k -1 h 4 k c v -1 2 s -1 h -1 k 2 n 2 d z 1 2 m -1 2 u c v -1 2 s h k -1 2 d z 1 2 m -1 2 u c v -1 2 s 2 d m -1 2 u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k 2 d m -1 2 u -1 2 d z 1 2 m -1 2 u c v -1 2 s 2 d m -1 2 u -1 1 2 Gamma 1 2 h k 2 -1 2 d z 1 2 m -1 2 u c v -1 2 s 2 4 d m -1 2 u -1 -1 c v -1 2 s 2 d z 1 2 m -1 2 u c v -1 2 s Gamma 1 2 h k 1 -1 2 d z 1 2 m -1 2 u c v -1 2 s 2 4 d m -1 2 u -1 -1 d m -1 2 u 2 n -1 m c 2 v -1 2 s 2 2 d m -1 2 u -1 d m -1 2 u 2 n h 0 k k 0 n -1 k -1 h 4 k c v -1 2 s -1 h -1 k 2 n c v -1 2 s -1 2 d m -1 2 u z 1 2 h k 2 d z 1 2 m -1 2 u c 2 s -1 v 2 d m -1 2 u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c v -1 2 s c v -1 2 s -1 2 d m -1 2 u z 1 2 Gamma 1 2 h k 1 2 d z 1 2 m -1 2 u c 2 s -1 v 2 4 d m -1 2 u -1 -1 2 d m -1 2 u 2 d z 1 2 m -1 2 u c 2 s -1 v 2 d m -1 2 u -1 1 2 Gamma 1 2 h k 2 2 d z 1 2 m -1 2 u c 2 s -1 v 2 4 d m -1 2 u -1 -1 m c 2 v -1 2 s 2 2 d m -1 2 u -1 d m -1 2 u 2 n h 0 k k 0 n -1 k -1 h 4 k c 2 s -1 v -1 h -1 k 2 n c 2 s -1 v -1 2 d m -1 2 u z 1 2 h k 2 d z 1 2 m -1 2 u c v -1 2 s 2 d m -1 2 u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c 2 s -1 v c 2 s -1 v -1 2 d m -1 2 u z 1 2 Gamma 1 2 h k 1 2 d z 1 2 m -1 2 u c v -1 2 s 2 4 d m -1 2 u -1 -1 2 d m -1 2 u 2 d z 1 2 m -1 2 u c v -1 2 s 2 d m -1 2 u -1 1 2 Gamma 1 2 h k 2 2 d z 1 2 m -1 2 u c v -1 2 s 2 4 d m -1 2 u -1 n m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18