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 Cos

 http://functions.wolfram.com/01.07.21.2536.01

 Input Form

 Integrate[z^n Sin[d z]^m Cos[c Sqrt[z] + g]^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 d^(-1 - 2 n) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[((-1)^k (2 k - m)^(-1 - 2 n) Binomial[m, k] ((-E^(I m Pi)) (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]))/ E^((1/2) I m Pi), {k, 0, Floor[(1/2) (-1 + m)]}] + (4 (-1)^n Binomial[m, m/2] (1 - Mod[m, 2]) Sum[((-2 s + v)^(-2 - 2 n) Binomial[v, s] (E^(2 I g (2 s - v)) Gamma[2 (1 + n), (-I) c (2 s - v) Sqrt[z]] + Gamma[2 (1 + n), I c (2 s - v) Sqrt[z]]))/E^(I g (2 s - v)), {s, 0, Floor[(1/2) (-1 + v)]}])/c^(2 (1 + n)) - Sum[(-1)^u Binomial[m, u] Sum[(Binomial[v, s] (((d^2 (m - 2 u)^2)^(-1 - 2 n) (((-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^((1/2) I (8 g s - 4 g v + (c^2 (-2 s + v)^2)/ (d (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}]))/ E^((I c^2 (-2 s + v)^2)/(4 d (m - 2 u))) + (1/(d m - 2 d u)^2) ((E^(I (g (4 s - 2 v) - (2 c s - c v)^2/(4 d m - 8 d u))) Sum[(-1)^(-h + k) 4^k (I (2 c s - c v))^(-h - k + 2 n) (I (c (2 s - v) + 2 (d m - 2 d u) Sqrt[z]))^(h + k) (-((I (c (2 s - v) + 2 (d m - 2 d u) Sqrt[z])^2)/(d m - 2 d u)))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((-2 c s + c v) (c (2 s - v) + 2 (d m - 2 d u) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (c (2 s - v) + 2 (d m - 2 d u) Sqrt[z])^2)/(4 d m - 8 d u))] + 2 I (d m - 2 d u) Sqrt[-((I (c (2 s - v) + 2 (d m - 2 d u) Sqrt[z])^2)/(d m - 2 d u))] Gamma[(1/2) (2 + h + k), -((I (c (2 s - v) + 2 (d m - 2 d u) Sqrt[z])^2)/(4 d m - 8 d u))]), {k, 0, n}, {h, 0, k}])/(I (d m - 2 d u))^ (2 n)) + (1/(d m - 2 d u)^2) (((-1)^m E^((I (2 c s - c v)^2)/ (4 d m - 8 d u)) Sum[(-1)^(-h + k) 4^k ((-I) (2 c s - c v))^ (-h - k + 2 n) ((-I) (c (2 s - v) + 2 (d m - 2 d u) Sqrt[z]))^(h + k) ((I (c (2 s - v) + 2 (d m - 2 d u) Sqrt[z])^2)/(d m - 2 d u))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((-2 c s + c v) (c (2 s - v) + 2 (d m - 2 d u) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (c (2 s - v) + 2 (d m - 2 d u) Sqrt[z])^2)/(4 d m - 8 d u)] - 2 I (d m - 2 d u) Sqrt[(I (c (2 s - v) + 2 (d m - 2 d u) Sqrt[z])^2)/(d m - 2 d u)] Gamma[(1/2) (2 + h + k), (I (c (2 s - v) + 2 (d m - 2 d u) Sqrt[z])^2)/(4 d m - 8 d u)]), {k, 0, n}, {h, 0, k}])/ ((-I) (d m - 2 d u))^(2 n))))/E^(I (2 g s - g v)), {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 ) cos v ( z c + g ) 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 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 - 1 2 m π ( 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]]]]] ( ( - d ( 2 k - m ) ) n Γ ( n + 1 , d ( 2 k - m ) z ) - m π ( d ( 2 k - m ) ) n Γ ( n + 1 , - d ( 2 k - m ) z ) ) + 4 ( - 1 ) n c - 2 ( 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) s = 0 v - 1 2 - g ( 2 s - v ) ( v - 2 s ) - 2 n - 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 g ( 2 s - v ) Γ ( 2 ( n + 1 ) , - c ( 2 s - v ) z ) + 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 - ( 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]]]]] ( - c 2 ( v - 2 s ) 2 4 d ( m - 2 u ) ( d 2 ( m - 2 u ) 2 ) - 2 n - 1 ( ( 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 1 2 ( c 2 ( v - 2 s ) 2 d ( m - 2 u ) + 8 g s - 4 g v ) ( 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 ( - ( 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]]]]] ( - 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 ) ) - 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 ) ) ) ) + 1 ( d m - 2 d u ) 2 ( ( - 1 ) m ( 2 c s - c v ) 2 4 d m - 8 d u ( - ( d m - 2 d u ) ) - 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - ( 2 c s - c v ) ) - h - k + 2 n ( - ( 2 z ( d m - 2 d u ) + c ( 2 s - v ) ) ) h + k ( ( 2 z ( d m - 2 d u ) + c ( 2 s - v ) ) 2 d m - 2 d 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 c s ) ( 2 z ( d m - 2 d u ) + c ( 2 s - v ) ) Γ ( 1 2 ( h + k + 1 ) , ( 2 z ( d m - 2 d u ) + c ( 2 s - v ) ) 2 4 d m - 8 d u ) - 2 ( d m - 2 d u ) ( 2 z ( d m - 2 d u ) + c ( 2 s - v ) ) 2 d m - 2 d u Γ ( 1 2 ( h + k + 2 ) , ( 2 z ( d m - 2 d u ) + c ( 2 s - v ) ) 2 4 d m - 8 d u ) ) ) + 1 ( d m - 2 d u ) 2 ( ( g ( 4 s - 2 v ) - ( 2 c s - c v ) 2 4 d m - 8 d u ) ( ( d m - 2 d u ) ) - 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( ( 2 c s - c v ) ) - h - k + 2 n ( ( 2 z ( d m - 2 d u ) + c ( 2 s - v ) ) ) h + k ( - ( 2 z ( d m - 2 d u ) + c ( 2 s - v ) ) 2 d m - 2 d 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 c s ) ( 2 z ( d m - 2 d u ) + c ( 2 s - v ) ) Γ ( 1 2 ( h + k + 1 ) , - ( 2 z ( d m - 2 d u ) + c ( 2 s - v ) ) 2 4 d m - 8 d u ) + 2 - ( 2 z ( d m - 2 d u ) + c ( 2 s - v ) ) 2 d m - 2 d u ( d m - 2 d u ) Γ ( 1 2 ( h + k + 2 ) , - ( 2 z ( d m - 2 d u ) + c ( 2 s - v ) ) 2 4 d m - 8 d u ) ) ) ) ) /; n m + v + Condition z z n d z m z 1 2 c g 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 d -2 n -1 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k -1 1 2 m 2 k -1 m -2 n -1 Binomial m k -1 d 2 k -1 m n Gamma n 1 d 2 k -1 m z -1 m d 2 k -1 m n Gamma n 1 -1 d 2 k -1 m z 4 -1 n c -2 n 1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 -1 g 2 s -1 v v -1 2 s -2 n -2 Binomial v s 2 g 2 s -1 v Gamma 2 n 1 -1 c 2 s -1 v z 1 2 Gamma 2 n 1 c 2 s -1 v z 1 2 -1 -1 m 4 -1 n u 0 m -1 2 -1 -1 u Binomial m u s 0 v -1 2 -1 -1 2 g s -1 g v Binomial v s -1 c 2 v -1 2 s 2 4 d m -1 2 u -1 d 2 m -1 2 u 2 -2 n -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 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 1 2 c 2 v -1 2 s 2 d m -1 2 u -1 8 g s -1 4 g v d m -1 2 u 2 n h 0 k k 0 n -1 k -1 h 4 k -1 c v -1 2 s -1 h -1 k 2 n -1 2 d z 1 2 m -1 2 u c v -1 2 s 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 -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 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 1 d m -1 2 d u 2 -1 -1 m 2 c s -1 c v 2 4 d m -1 8 d u -1 -1 d m -1 2 d u -2 n h 0 k k 0 n -1 k -1 h 4 k -1 2 c s -1 c v -1 h -1 k 2 n -1 2 z 1 2 d m -1 2 d u c 2 s -1 v h k 2 z 1 2 d m -1 2 d u c 2 s -1 v 2 d m -1 2 d u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c v -1 2 c s 2 z 1 2 d m -1 2 d u c 2 s -1 v Gamma 1 2 h k 1 2 z 1 2 d m -1 2 d u c 2 s -1 v 2 4 d m -1 8 d u -1 -1 2 d m -1 2 d u 2 z 1 2 d m -1 2 d u c 2 s -1 v 2 d m -1 2 d u -1 1 2 Gamma 1 2 h k 2 2 z 1 2 d m -1 2 d u c 2 s -1 v 2 4 d m -1 8 d u -1 1 d m -1 2 d u 2 -1 g 4 s -1 2 v -1 2 c s -1 c v 2 4 d m -1 8 d u -1 d m -1 2 d u -2 n h 0 k k 0 n -1 k -1 h 4 k 2 c s -1 c v -1 h -1 k 2 n 2 z 1 2 d m -1 2 d u c 2 s -1 v h k -1 2 z 1 2 d m -1 2 d u c 2 s -1 v 2 d m -1 2 d u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c v -1 2 c s 2 z 1 2 d m -1 2 d u c 2 s -1 v Gamma 1 2 h k 1 -1 2 z 1 2 d m -1 2 d u c 2 s -1 v 2 4 d m -1 8 d u -1 2 -1 2 z 1 2 d m -1 2 d u c 2 s -1 v 2 d m -1 2 d u -1 1 2 d m -1 2 d u Gamma 1 2 h k 2 -1 2 z 1 2 d m -1 2 d u c 2 s -1 v 2 4 d m -1 8 d 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