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 Cosh

 http://functions.wolfram.com/01.20.21.3981.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18