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 Cosh

 http://functions.wolfram.com/01.20.21.1439.01

 Input Form

 Integrate[z^n E^(p z) Cos[b Sqrt[z]]^m Cosh[c z], z] == 2^(-2 - m) (2 Binomial[m, m/2] ((c - p)^(-1 - n) Gamma[1 + n, (c - p) z] + (-c - p)^(-1 - n) Gamma[1 + n, (-(c + p)) z]) (-1 + Mod[m, 2]) + Sum[Binomial[m, u] ((E^((b^2 (m - 2 u)^2)/(4 (-c + p))) Sum[(-1)^(-h + k) 4^k (I b (m - 2 u))^(-h - k + 2 n) ((b (m - 2 u) + 2 I (c - p) Sqrt[z])^2/(-c + p))^ ((1/2) (-1 - h - k)) (I b (m - 2 u) + 2 (-c + p) Sqrt[z])^ (h + k) Binomial[k, h] Binomial[n, k] (I b (m - 2 u) (I b (m - 2 u) + 2 (-c + p) Sqrt[z]) Gamma[(1/2) (1 + h + k), (b (m - 2 u) + 2 I (c - p) Sqrt[z])^2/ (4 (-c + p))] + 2 (-c + p) Sqrt[(b (m - 2 u) + 2 I (c - p) Sqrt[z])^2/(-c + p)] Gamma[(1/2) (2 + h + k), (b (m - 2 u) + 2 I (c - p) Sqrt[z])^2/(4 (-c + p))]), {k, 0, n}, {h, 0, k}])/(-c + p)^(2 (1 + n)) + (E^((b^2 (m - 2 u)^2)/(4 (-c + p))) Sum[(-1)^(-h + k) 4^k ((-I) b (m - 2 u))^(-h - k + 2 n) ((b (m - 2 u) + 2 I (-c + p) Sqrt[z])^2/(-c + p))^ ((1/2) (-1 - h - k)) ((-I) b (m - 2 u) + 2 (-c + p) Sqrt[z])^ (h + k) Binomial[k, h] Binomial[n, k] ((-I) b (m - 2 u) ((-I) b (m - 2 u) + 2 (-c + p) Sqrt[z]) Gamma[(1/2) (1 + h + k), (b (m - 2 u) + 2 I (-c + p) Sqrt[z])^2/ (4 (-c + p))] + 2 (-c + p) Sqrt[(b (m - 2 u) + 2 I (-c + p) Sqrt[z])^2/(-c + p)] Gamma[(1/2) (2 + h + k), (b (m - 2 u) + 2 I (-c + p) Sqrt[z])^2/(4 (-c + p))]), {k, 0, n}, {h, 0, k}])/(-c + p)^(2 (1 + n)) + (E^((b^2 (m - 2 u)^2)/(4 (c + p))) (Sum[(-1)^(-h + k) 4^k (I b (m - 2 u))^(-h - k + 2 n) ((b (m - 2 u) - 2 I (c + p) Sqrt[z])^2/(c + p))^((1/2) (-1 - h - k)) (I b (m - 2 u) + 2 (c + p) Sqrt[z])^(h + k) Binomial[k, h] Binomial[n, k] (I b (m - 2 u) (I b (m - 2 u) + 2 (c + p) Sqrt[z]) Gamma[(1/2) (1 + h + k), (b (m - 2 u) - 2 I (c + p) Sqrt[z])^2/(4 (c + p))] + 2 (c + p) Sqrt[(b (m - 2 u) - 2 I (c + p) Sqrt[z])^2/(c + p)] Gamma[(1/2) (2 + h + k), (b (m - 2 u) - 2 I (c + p) Sqrt[z])^2/ (4 (c + p))]), {k, 0, n}, {h, 0, k}] + Sum[(-1)^(-h + k) 4^k ((-I) b (m - 2 u))^(-h - k + 2 n) ((b (m - 2 u) + 2 I (c + p) Sqrt[z])^2/(c + p))^((1/2) (-1 - h - k)) ((-I) b (m - 2 u) + 2 (c + p) Sqrt[z])^(h + k) Binomial[k, h] Binomial[n, k] ((-I) b (m - 2 u) ((-I) b (m - 2 u) + 2 (c + p) Sqrt[z]) Gamma[(1/2) (1 + h + k), (b (m - 2 u) + 2 I (c + p) Sqrt[z])^2/(4 (c + p))] + 2 (c + p) Sqrt[(b (m - 2 u) + 2 I (c + p) Sqrt[z])^2/(c + p)] Gamma[(1/2) (2 + h + k), (b (m - 2 u) + 2 I (c + p) Sqrt[z])^2/ (4 (c + p))]), {k, 0, n}, {h, 0, k}]))/(c + p)^(2 (1 + n))), {u, 0, Floor[(1/2) (-1 + m)]}]/4^n) /; Element[m, Integers] && m > 0 && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n p z cos m ( b z ) cosh ( c z ) z 2 - m - 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]]]]] ( Γ ( n + 1 , - ( c + p ) z ) ( - c - p ) - n - 1 + ( c - p ) - n - 1 Γ ( n + 1 , ( c - p ) z ) ) ( m mod 2 \$CellContext`m 2 - 1 ) + 4 - n u = 0 m - 1 2 ( m u ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["u", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b 2 ( m - 2 u ) 2 4 ( p - c ) ( p - c ) - 2 ( n + 1 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b ( m - 2 u ) ) - h - k + 2 n ( ( 2 z ( c - p ) + b ( m - 2 u ) ) 2 p - c ) 1 2 ( - h - k - 1 ) ( 2 z ( p - c ) + 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 ) ( 2 z ( p - c ) + b ( m - 2 u ) ) Γ ( 1 2 ( h + k + 1 ) , ( 2 z ( c - p ) + b ( m - 2 u ) ) 2 4 ( p - c ) ) + 2 ( 2 z ( c - p ) + b ( m - 2 u ) ) 2 p - c ( p - c ) Γ ( 1 2 ( h + k + 2 ) , ( 2 z ( c - p ) + b ( m - 2 u ) ) 2 4 ( p - c ) ) ) + b 2 ( m - 2 u ) 2 4 ( p - c ) ( p - c ) - 2 ( n + 1 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b ( m - 2 u ) ) - h - k + 2 n ( ( 2 z ( p - c ) + b ( m - 2 u ) ) 2 p - c ) 1 2 ( - h - k - 1 ) ( 2 ( p - c ) 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]]]]] ( 2 ( p - c ) ( 2 z ( p - c ) + b ( m - 2 u ) ) 2 p - c Γ ( 1 2 ( h + k + 2 ) , ( 2 z ( p - c ) + b ( m - 2 u ) ) 2 4 ( p - c ) ) - b ( m - 2 u ) ( 2 ( p - c ) z - b ( m - 2 u ) ) Γ ( 1 2 ( h + k + 1 ) , ( 2 z ( p - c ) + b ( m - 2 u ) ) 2 4 ( p - c ) ) ) + b 2 ( m - 2 u ) 2 4 ( c + p ) ( c + p ) - 2 ( n + 1 ) ( k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b ( m - 2 u ) ) - h - k + 2 n ( ( 2 z ( c + p ) + b ( m - 2 u ) ) 2 c + p ) 1 2 ( - h - k - 1 ) ( 2 ( c + p ) 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]]]]] ( 2 ( c + p ) ( 2 z ( c + p ) + b ( m - 2 u ) ) 2 c + p Γ ( 1 2 ( h + k + 2 ) , ( 2 z ( c + p ) + b ( m - 2 u ) ) 2 4 ( c + p ) ) - b ( m - 2 u ) ( 2 ( c + p ) z - b ( m - 2 u ) ) Γ ( 1 2 ( h + k + 1 ) , ( 2 z ( c + p ) + b ( m - 2 u ) ) 2 4 ( c + p ) ) ) + 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 + p ) z ) 2 c + p ) 1 2 ( - h - k - 1 ) ( 2 z ( c + p ) + 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 ) ( 2 z ( c + p ) + b ( m - 2 u ) ) Γ ( 1 2 ( h + k + 1 ) , ( b ( m - 2 u ) - 2 ( c + p ) z ) 2 4 ( c + p ) ) + 2 ( b ( m - 2 u ) - 2 ( c + p ) z ) 2 c + p ( c + p ) Γ ( 1 2 ( h + k + 2 ) , ( b ( m - 2 u ) - 2 ( c + p ) z ) 2 4 ( c + p ) ) ) ) ) ) /; m + n Condition z z n p z b z 1 2 m c z 2 -1 m -2 2 Binomial m m 2 -1 Gamma n 1 -1 c p z -1 c -1 p -1 n -1 c -1 p -1 n -1 Gamma n 1 c -1 p z \$CellContext`m 2 -1 4 -1 n u 0 m -1 2 -1 Binomial m u b 2 m -1 2 u 2 4 p -1 c -1 p -1 c -2 n 1 h 0 k k 0 n -1 k -1 h 4 k b m -1 2 u -1 h -1 k 2 n 2 z 1 2 c -1 p b m -1 2 u 2 p -1 c -1 1 2 -1 h -1 k -1 2 z 1 2 p -1 c b m -1 2 u h k Binomial k h Binomial n k b m -1 2 u 2 z 1 2 p -1 c b m -1 2 u Gamma 1 2 h k 1 2 z 1 2 c -1 p b m -1 2 u 2 4 p -1 c -1 2 2 z 1 2 c -1 p b m -1 2 u 2 p -1 c -1 1 2 p -1 c Gamma 1 2 h k 2 2 z 1 2 c -1 p b m -1 2 u 2 4 p -1 c -1 b 2 m -1 2 u 2 4 p -1 c -1 p -1 c -2 n 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 2 z 1 2 p -1 c b m -1 2 u 2 p -1 c -1 1 2 -1 h -1 k -1 2 p -1 c z 1 2 -1 b m -1 2 u h k Binomial k h Binomial n k 2 p -1 c 2 z 1 2 p -1 c b m -1 2 u 2 p -1 c -1 1 2 Gamma 1 2 h k 2 2 z 1 2 p -1 c b m -1 2 u 2 4 p -1 c -1 -1 b m -1 2 u 2 p -1 c z 1 2 -1 b m -1 2 u Gamma 1 2 h k 1 2 z 1 2 p -1 c b m -1 2 u 2 4 p -1 c -1 b 2 m -1 2 u 2 4 c p -1 c p -2 n 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 2 z 1 2 c p b m -1 2 u 2 c p -1 1 2 -1 h -1 k -1 2 c p z 1 2 -1 b m -1 2 u h k Binomial k h Binomial n k 2 c p 2 z 1 2 c p b m -1 2 u 2 c p -1 1 2 Gamma 1 2 h k 2 2 z 1 2 c p b m -1 2 u 2 4 c p -1 -1 b m -1 2 u 2 c p z 1 2 -1 b m -1 2 u Gamma 1 2 h k 1 2 z 1 2 c p b m -1 2 u 2 4 c p -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 -1 2 c p z 1 2 2 c p -1 1 2 -1 h -1 k -1 2 z 1 2 c p b m -1 2 u h k Binomial k h Binomial n k b m -1 2 u 2 z 1 2 c p b m -1 2 u Gamma 1 2 h k 1 b m -1 2 u -1 2 c p z 1 2 2 4 c p -1 2 b m -1 2 u -1 2 c p z 1 2 2 c p -1 1 2 c p Gamma 1 2 h k 2 b m -1 2 u -1 2 c p z 1 2 2 4 c p -1 m SuperPlus n [/itex]

 Rule Form

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

 2002-12-18