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/01.07.21.2506.01

 Input Form

 Integrate[z^n Sin[b Sqrt[z]]^m Cos[c 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[m, m/2] (-1 + Mod[m, 2]) Sum[Binomial[v, s] (((-I) c (2 s - v))^(-1 - n) Gamma[1 + n, (-I) c (2 s - v) z] + (I c (2 s - v))^(-1 - n) Gamma[1 + n, I c (2 s - v) z]), {s, 0, Floor[(1/2) (-1 + v)]}] + (4 Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[(-1)^u Binomial[m, u] (Gamma[2 (1 + n), (-I) b (m - 2 u) Sqrt[z]]/((-I) b (m - 2 u))^ (2 (1 + n)) + ((-1)^m Gamma[2 (1 + n), I b (m - 2 u) Sqrt[z]])/ (I b (m - 2 u))^(2 (1 + n))), {u, 0, Floor[(1/2) (-1 + m)]}])/I^m + Sum[(-1)^u Binomial[m, u] Sum[(-E^(-((I b^2 (m - 2 u)^2)/(c (8 s - 4 v))))) (c^2 (-2 s + v)^2)^(-1 - 2 n) Binomial[v, s] ((I c (-2 s + v))^(2 n) Sum[I (-1)^(-h + k) 4^k (I b (m - 2 u))^ (-h - k + 2 n) (I (b (m - 2 u) + 2 c (2 s - v) Sqrt[z]))^ (h + k) (-((I (b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^2)/ (c (2 s - v))))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (I b (m - 2 u) (b (m - 2 u) + 2 c (2 s - v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^2)/ (c (8 s - 4 v)))] + 2 c (2 s - v) Sqrt[ -((I (b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^2)/(c (2 s - v)))] Gamma[(1/2) (2 + h + k), -((I (b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^2)/(c (8 s - 4 v)))]), {k, 0, n}, {h, 0, k}] + (-1)^m E^((I b^2 (m - 2 u)^2)/(4 c s - 2 c v)) (I c (2 s - v))^ (2 n) Sum[I (-1)^(-h + k) 4^k ((-I) b (m - 2 u))^(-h - k + 2 n) ((I (b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^2)/(c (2 s - v)))^ ((1/2) (-1 - h - k)) (I ((-b) (m - 2 u) + 2 c (-2 s + v) Sqrt[z]))^(h + k) Binomial[k, h] Binomial[n, k] (I b (m - 2 u) (b (m - 2 u) + 2 c (2 s - v) Sqrt[z]) Gamma[ (1/2) (1 + h + k), (I (b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^ 2)/(c (8 s - 4 v))] + 2 c (-2 s + v) Sqrt[ (I (b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^2)/(c (2 s - v))] Gamma[(1/2) (2 + h + k), (I (b (m - 2 u) + 2 c (2 s - v) Sqrt[z])^2)/(c (8 s - 4 v))]), {k, 0, n}, {h, 0, k}] + (-1)^m (I c (-2 s + v))^(2 n) Sum[I (-1)^(-h + k) 4^k ((-I) b (m - 2 u))^(-h - k + 2 n) (I ((-b) (m - 2 u) + 2 c (2 s - v) Sqrt[z]))^(h + k) (-((I (b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^2)/(c (2 s - v))))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (I b (m - 2 u) (b (m - 2 u) + 2 c (-2 s + v) Sqrt[z]) Gamma[ (1/2) (1 + h + k), -((I (b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^2)/(c (8 s - 4 v)))] + 2 c (2 s - v) Sqrt[ -((I (b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^2)/ (c (2 s - v)))] Gamma[(1/2) (2 + h + k), -((I (b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^2)/ (c (8 s - 4 v)))]), {k, 0, n}, {h, 0, k}] + E^((I b^2 (m - 2 u)^2)/(4 c s - 2 c v)) (I c (2 s - v))^(2 n) Sum[I (-1)^(-h + k) 4^k (I b (m - 2 u))^(-h - k + 2 n) (I (b (m - 2 u) + 2 c (-2 s + v) Sqrt[z]))^(h + k) ((I (b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^2)/(c (2 s - v)))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (I b (m - 2 u) (b (m - 2 u) + 2 c (-2 s + v) Sqrt[z]) Gamma[ (1/2) (1 + h + k), (I (b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^ 2)/(c (8 s - 4 v))] + 2 c (-2 s + v) Sqrt[ (I (b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^2)/(c (2 s - v))] Gamma[(1/2) (2 + h + k), (I (b (m - 2 u) + 2 c (-2 s + v) Sqrt[z])^2)/(c (8 s - 4 v))]), {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 ( b z ) 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 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 , - c ( 2 s - v ) z ) ( - c ( 2 s - v ) ) - n - 1 + ( c ( 2 s - v ) ) - n - 1 Γ ( n + 1 , c ( 2 s - v ) z ) ) + 4 - 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]]]]] ( Γ ( 2 ( n + 1 ) , - b ( m - 2 u ) z ) ( - b ( m - 2 u ) ) - 2 ( n + 1 ) + ( - 1 ) m ( b ( m - 2 u ) ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , b ( m - 2 u ) 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 - - b 2 ( m - 2 u ) 2 c ( 8 s - 4 v ) ( c 2 ( v - 2 s ) 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]]]]] ( ( - 1 ) m b 2 ( m - 2 u ) 2 4 c s - 2 c v ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b ( m - 2 u ) ) - h - k + 2 n ( ( b ( m - 2 u ) + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) ) 1 2 ( - h - k - 1 ) ( ( 2 c ( v - 2 s ) z - b ( m - 2 u ) ) ) h + k ( 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 ( m - 2 u ) ( b ( m - 2 u ) + 2 c ( 2 s - v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b ( m - 2 u ) + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( b ( m - 2 u ) + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , ( b ( m - 2 u ) + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) ) ) ( c ( 2 s - v ) ) 2 n + b 2 ( m - 2 u ) 2 4 c s - 2 c v ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b ( m - 2 u ) ) - h - k + 2 n ( ( b ( m - 2 u ) + 2 c ( v - 2 s ) z ) ) h + k ( ( b ( m - 2 u ) + 2 c ( v - 2 s ) z ) 2 c ( 2 s - 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 ( m - 2 u ) ( b ( m - 2 u ) + 2 c ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b ( m - 2 u ) + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( b ( m - 2 u ) + 2 c ( v - 2 s ) z ) 2 c ( 2 s - v ) ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , ( b ( m - 2 u ) + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) ) ) ( c ( 2 s - v ) ) 2 n + ( c ( v - 2 s ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b ( m - 2 u ) ) - h - k + 2 n ( ( b ( m - 2 u ) + 2 c ( 2 s - v ) z ) ) h + k ( - ( b ( m - 2 u ) + 2 c ( 2 s - v ) z ) 2 c ( 2 s - 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 ( m - 2 u ) ( b ( m - 2 u ) + 2 c ( 2 s - v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b ( m - 2 u ) + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( 2 s - v ) - ( b ( m - 2 u ) + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , - ( b ( m - 2 u ) + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) ) + ( - 1 ) m ( c ( v - 2 s ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b ( m - 2 u ) ) - h - k + 2 n ( ( 2 c ( 2 s - v ) z - b ( m - 2 u ) ) ) h + k ( - ( b ( m - 2 u ) + 2 c ( v - 2 s ) z ) 2 c ( 2 s - 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 ( m - 2 u ) ( b ( m - 2 u ) + 2 c ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b ( m - 2 u ) + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( 2 s - v ) - ( b ( m - 2 u ) + 2 c ( v - 2 s ) z ) 2 c ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , - ( b ( m - 2 u ) + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) ) ) ) /; n m + v + Condition z z n b z 1 2 m c z 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 Binomial m m 2 -1 \$CellContext`m 2 -1 s 0 v -1 2 -1 Binomial v s Gamma n 1 -1 c 2 s -1 v z -1 c 2 s -1 v -1 n -1 c 2 s -1 v -1 n -1 Gamma n 1 c 2 s -1 v z 4 -1 m Binomial v v 2 -1 \$CellContext`v 2 -1 u 0 m -1 2 -1 -1 u Binomial m u Gamma 2 n 1 -1 b m -1 2 u z 1 2 -1 b m -1 2 u -2 n 1 -1 m b m -1 2 u -2 n 1 Gamma 2 n 1 b m -1 2 u 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 b 2 m -1 2 u 2 c 8 s -1 4 v -1 c 2 v -1 2 s 2 -2 n -1 Binomial v s -1 m b 2 m -1 2 u 2 4 c s -1 2 c v -1 h 0 k k 0 n -1 k -1 h 4 k -1 b m -1 2 u -1 h -1 k 2 n b m -1 2 u 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 -1 h -1 k -1 2 c v -1 2 s z 1 2 -1 b m -1 2 u h k Binomial k h Binomial n k b m -1 2 u b m -1 2 u 2 c 2 s -1 v z 1 2 Gamma 1 2 h k 1 b m -1 2 u 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 2 c b m -1 2 u 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 v -1 2 s Gamma 1 2 h k 2 b m -1 2 u 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 c 2 s -1 v 2 n b 2 m -1 2 u 2 4 c s -1 2 c v -1 h 0 k k 0 n -1 k -1 h 4 k b m -1 2 u -1 h -1 k 2 n b m -1 2 u 2 c v -1 2 s z 1 2 h k b m -1 2 u 2 c v -1 2 s z 1 2 2 c 2 s -1 v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b m -1 2 u b m -1 2 u 2 c v -1 2 s z 1 2 Gamma 1 2 h k 1 b m -1 2 u 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 2 c b m -1 2 u 2 c v -1 2 s z 1 2 2 c 2 s -1 v -1 1 2 v -1 2 s Gamma 1 2 h k 2 b m -1 2 u 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 c 2 s -1 v 2 n c v -1 2 s 2 n h 0 k k 0 n -1 k -1 h 4 k b m -1 2 u -1 h -1 k 2 n b m -1 2 u 2 c 2 s -1 v z 1 2 h k -1 b m -1 2 u 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b m -1 2 u b m -1 2 u 2 c 2 s -1 v z 1 2 Gamma 1 2 h k 1 -1 b m -1 2 u 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 2 c 2 s -1 v -1 b m -1 2 u 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 Gamma 1 2 h k 2 -1 b m -1 2 u 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 -1 m c v -1 2 s 2 n h 0 k k 0 n -1 k -1 h 4 k -1 b m -1 2 u -1 h -1 k 2 n 2 c 2 s -1 v z 1 2 -1 b m -1 2 u h k -1 b m -1 2 u 2 c v -1 2 s z 1 2 2 c 2 s -1 v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b m -1 2 u b m -1 2 u 2 c v -1 2 s z 1 2 Gamma 1 2 h k 1 -1 b m -1 2 u 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 2 c 2 s -1 v -1 b m -1 2 u 2 c v -1 2 s z 1 2 2 c 2 s -1 v -1 1 2 Gamma 1 2 h k 2 -1 b m -1 2 u 2 c v -1 2 s z 1 2 2 c 8 s -1 4 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