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 Cosh

 http://functions.wolfram.com/01.20.21.3979.01

 Input Form

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

 Rule Form

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

 2002-12-18