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 Cosh

 http://functions.wolfram.com/01.20.21.1415.01

 Input Form

 Integrate[z^n E^(p Sqrt[z]) Cos[b z] Cosh[c Sqrt[z]], z] == (-1)^n 2^(-3 - 2 n) b^(-2 - 2 n) ((-E^((I (c + p)^2)/(4 b))) Sum[(-1)^(-h + k) 4^k (c + p)^(-h - k + 2 n) (c + p + 2 I b Sqrt[z])^ (h + k) ((I (c + p + 2 I b Sqrt[z])^2)/b)^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((c + p) (c + p + 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (I c + I p - 2 b Sqrt[z])^2)/(4 b))] + 2 I b Sqrt[(I (c + p + 2 I b Sqrt[z])^2)/b] Gamma[(1/2) (2 + h + k), -((I (I c + I p - 2 b Sqrt[z])^2)/(4 b))]), {k, 0, n}, {h, 0, k}] - Sum[(-1)^(-h + k) 4^k (-c + p)^(-h - k + 2 n) (-c + p - 2 I b Sqrt[z])^ (h + k) (-((I (-c + p - 2 I b Sqrt[z])^2)/b))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((-c + p) (-c + p - 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (-c + p - 2 I b Sqrt[z])^2)/(4 b))] - 2 I b Sqrt[-((I (-c + p - 2 I b Sqrt[z])^2)/b)] Gamma[(1/2) (2 + h + k), -((I (-c + p - 2 I b Sqrt[z])^2)/(4 b))]), {k, 0, n}, {h, 0, k}]/E^((I (-c + p)^2)/(4 b)) - E^((I (-c + p)^2)/(4 b)) Sum[(-1)^(-h + k) 4^k (-c + p)^(-h - k + 2 n) (-c + p + 2 I b Sqrt[z])^(h + k) ((I (-c + p + 2 I b Sqrt[z])^2)/b)^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((-c + p) (-c + p + 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I (I c - I p + 2 b Sqrt[z])^2)/(4 b))] + 2 I b Sqrt[(I (-c + p + 2 I b Sqrt[z])^2)/b] Gamma[(1/2) (2 + h + k), -((I (I c - I p + 2 b Sqrt[z])^2)/(4 b))]), {k, 0, n}, {h, 0, k}] - Sum[(-1)^(-h + k) 4^k (c + p)^(-h - k + 2 n) (c + p - 2 I b Sqrt[z])^ (h + k) ((I (I c + I p + 2 b Sqrt[z])^2)/b)^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((c + p) (c + p - 2 I b Sqrt[z]) Gamma[(1/2) (1 + h + k), (I (I c + I p + 2 b Sqrt[z])^2)/(4 b)] - 2 I b Sqrt[(I (I c + I p + 2 b Sqrt[z])^2)/b] Gamma[(1/2) (2 + h + k), (I (I c + I p + 2 b Sqrt[z])^2)/(4 b)]), {k, 0, n}, {h, 0, k}]/E^((I (c + p)^2)/(4 b))) /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n p z cos ( b z ) cosh ( c z ) z ( - 1 ) n 2 - 2 n - 3 b - 2 n - 2 ( - - ( c + p ) 2 4 b k = 0 n h = 0 k ( - 1 ) k - h 4 k ( c + p ) - h - k + 2 n ( c + p - 2 b z ) h + k ( ( 2 z b + c + p ) 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 + p ) ( c + p - 2 b z ) Γ ( 1 2 ( h + k + 1 ) , ( 2 z b + c + p ) 2 4 b ) - 2 b ( 2 z b + c + p ) 2 b Γ ( 1 2 ( h + k + 2 ) , ( 2 z b + c + p ) 2 4 b ) ) - ( p - c ) 2 4 b k = 0 n h = 0 k ( - 1 ) k - h 4 k ( p - c ) - h - k + 2 n ( - c + p + 2 b z ) h + k ( ( - c + p + 2 b z ) 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]]]]] ( ( p - c ) ( - c + p + 2 b z ) Γ ( 1 2 ( h + k + 1 ) , - ( 2 z b + c - p ) 2 4 b ) + 2 ( - c + p + 2 b z ) 2 b b Γ ( 1 2 ( h + k + 2 ) , - ( 2 z b + c - p ) 2 4 b ) ) - ( c + p ) 2 4 b k = 0 n h = 0 k ( - 1 ) k - h 4 k ( c + p ) - h - k + 2 n ( c + p + 2 b z ) h + k ( ( c + p + 2 b z ) 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 + p ) ( c + p + 2 b z ) Γ ( 1 2 ( h + k + 1 ) , - ( - 2 z b + c + p ) 2 4 b ) + 2 ( c + p + 2 b z ) 2 b b Γ ( 1 2 ( h + k + 2 ) , - ( - 2 z b + c + p ) 2 4 b ) ) - - ( p - c ) 2 4 b k = 0 n h = 0 k ( - 1 ) k - h 4 k ( p - c ) - h - k + 2 n ( - c + p - 2 b z ) h + k ( - ( - c + p - 2 b z ) 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]]]]] ( ( p - c ) ( - c + p - 2 b z ) Γ ( 1 2 ( h + k + 1 ) , - ( - c + p - 2 b z ) 2 4 b ) - 2 b - ( - c + p - 2 b z ) 2 b Γ ( 1 2 ( h + k + 2 ) , - ( - c + p - 2 b z ) 2 4 b ) ) ) /; n Condition z z n p z 1 2 b z c z 1 2 -1 n 2 -2 n -3 b -2 n -2 -1 -1 c p 2 4 b -1 h 0 k k 0 n -1 k -1 h 4 k c p -1 h -1 k 2 n c p -1 2 b z 1 2 h k 2 z 1 2 b c p 2 b -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c p c p -1 2 b z 1 2 Gamma 1 2 h k 1 2 z 1 2 b c p 2 4 b -1 -1 2 b 2 z 1 2 b c p 2 b -1 1 2 Gamma 1 2 h k 2 2 z 1 2 b c p 2 4 b -1 -1 p -1 c 2 4 b -1 h 0 k k 0 n -1 k -1 h 4 k p -1 c -1 h -1 k 2 n -1 c p 2 b z 1 2 h k -1 c p 2 b z 1 2 2 b -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p -1 c -1 c p 2 b z 1 2 Gamma 1 2 h k 1 -1 2 z 1 2 b c -1 p 2 4 b -1 2 -1 c p 2 b z 1 2 2 b -1 1 2 b Gamma 1 2 h k 2 -1 2 z 1 2 b c -1 p 2 4 b -1 -1 c p 2 4 b -1 h 0 k k 0 n -1 k -1 h 4 k c p -1 h -1 k 2 n c p 2 b z 1 2 h k c p 2 b z 1 2 2 b -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k c p c p 2 b z 1 2 Gamma 1 2 h k 1 -1 -2 z 1 2 b c p 2 4 b -1 2 c p 2 b z 1 2 2 b -1 1 2 b Gamma 1 2 h k 2 -1 -2 z 1 2 b c p 2 4 b -1 -1 -1 p -1 c 2 4 b -1 h 0 k k 0 n -1 k -1 h 4 k p -1 c -1 h -1 k 2 n -1 c p -1 2 b z 1 2 h k -1 -1 c p -1 2 b z 1 2 2 b -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p -1 c -1 c p -1 2 b z 1 2 Gamma 1 2 h k 1 -1 -1 c p -1 2 b z 1 2 2 4 b -1 -1 2 b -1 -1 c p -1 2 b z 1 2 2 b -1 1 2 Gamma 1 2 h k 2 -1 -1 c p -1 2 b z 1 2 2 4 b -1 n [/itex]

 Rule Form

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

 2002-12-18