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 Cosh

 http://functions.wolfram.com/01.20.21.3973.01

 Input Form

 Integrate[z^n E^(p Sqrt[z]) Cos[b Sqrt[z]] Cosh[c z]^v, z] == 2^(-2 - v) (4 Binomial[v, v/2] (Gamma[2 (1 + n), ((-I) b - p) Sqrt[z]]/ ((-I) b - p)^(2 (1 + n)) + Gamma[2 (1 + n), (I b - p) Sqrt[z]]/ (I b - p)^(2 (1 + n))) (-1 + Mod[v, 2]) + Sum[(Binomial[v, s] (E^((b + I p)^2/(c (8 s - 4 v))) (c (-2 s + v))^(2 n) Sum[(-1)^(-h + k) 4^k ((-I) b + p)^(-h - k + 2 n) ((b + I (p + 2 c (2 s - v) Sqrt[z]))^2/(c (2 s - v)))^ ((1/2) (-1 - h - k)) ((-I) b + p + 2 c (2 s - v) Sqrt[z])^ (h + k) Binomial[k, h] Binomial[n, k] (((-I) b + p) ((-I) b + p + 2 c (2 s - v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (b + I (p + 2 c (2 s - v) Sqrt[z]))^2/ (c (8 s - 4 v))] + 2 c (2 s - v) Sqrt[(b + I (p + 2 c (2 s - v) Sqrt[z]))^2/(c (2 s - v))] Gamma[(1/2) (2 + h + k), (b + I (p + 2 c (2 s - v) Sqrt[z]))^2/ (c (8 s - 4 v))]), {k, 0, n}, {h, 0, k}] + E^((I b + p)^2/(c (8 s - 4 v))) (E^((b - I p)^2/(4 c s - 2 c v)) (c (-2 s + v))^(2 n) Sum[(-1)^(-h + k) 4^k (I b + p)^(-h - k + 2 n) (I b + p + 2 c (2 s - v) Sqrt[z])^(h + k) (-((I b + p + 2 c (2 s - v) Sqrt[z])^2/(c (2 s - v))))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((I b + p) (I b + p + 2 c (2 s - v) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((I b + p + 2 c (2 s - v) Sqrt[z])^2/(c (8 s - 4 v)))] + 2 c (2 s - v) Sqrt[-((I b + p + 2 c (2 s - v) Sqrt[z])^2/ (c (2 s - v)))] Gamma[(1/2) (2 + h + k), -((I b + p + 2 c (2 s - v) Sqrt[z])^2/(c (8 s - 4 v)))]), {k, 0, n}, {h, 0, k}] + (c (2 s - v))^(2 n) (Sum[(-1)^(-h + k) 4^k ((-I) b + p)^(-h - k + 2 n) ((-I) b + p + 2 c (-2 s + v) Sqrt[z])^(h + k) (-(((-I) b + p + 2 c (-2 s + v) Sqrt[z])^2/(c (-2 s + v))))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (((-I) b + p) ((-I) b + p + 2 c (-2 s + v) Sqrt[z]) Gamma[(1/2) (1 + h + k), ((-I) b + p + 2 c (-2 s + v) Sqrt[z])^2/(c (8 s - 4 v))] + 2 c (-2 s + v) Sqrt[-(((-I) b + p + 2 c (-2 s + v) Sqrt[z])^2/ (c (-2 s + v)))] Gamma[(1/2) (2 + h + k), ((-I) b + p + 2 c (-2 s + v) Sqrt[z])^2/(c (8 s - 4 v))]), {k, 0, n}, {h, 0, k}]/E^((I b p)/(c (2 s - v))) + Sum[(-1)^(-h + k) 4^k (I b + p)^(-h - k + 2 n) (I b + p + 2 c (-2 s + v) Sqrt[z])^(h + k) (-((I b + p + 2 c (-2 s + v) Sqrt[z])^2/(c (-2 s + v))))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] ((I b + p) (I b + p + 2 c (-2 s + v) Sqrt[z]) Gamma[(1/2) (1 + h + k), (I b + p + 2 c (-2 s + v) Sqrt[z])^2/(c (8 s - 4 v))] + 2 c (-2 s + v) Sqrt[-((I b + p + 2 c (-2 s + v) Sqrt[z])^2/ (c (-2 s + v)))] Gamma[(1/2) (2 + h + k), (I b + p + 2 c (-2 s + v) Sqrt[z])^2/(c (8 s - 4 v))]), {k, 0, n}, {h, 0, k}]))))/((c (2 s - v))^(2 n) (c (-2 s + v))^(2 (1 + n))), {s, 0, Floor[(1/2) (-1 + v)]}]/4^n) /; Element[v, Integers] && v > 0 && Element[n, Integers] && n >= 0

 Standard Form

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RowBox[List[RowBox[List["8", " ", "s"]], "-", RowBox[List["4", " ", "v"]]]], ")"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], ")"]]]]]], ")"]]]]]], ")"]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n p z cos ( b z ) cosh v ( c z ) z 2 - v - 2 ( 4 ( 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 ( 2 ( n + 1 ) , ( - b - p ) z ) ( - b - p ) - 2 ( n + 1 ) + ( b - p ) - 2 ( n + 1 ) Γ ( 2 ( n + 1 ) , ( b - p ) z ) ) ( v mod 2 \$CellContext`v 2 - 1 ) + 4 - n s = 0 v - 1 2 ( c ( 2 s - v ) ) - 2 n ( c ( v - 2 s ) ) - 2 ( n + 1 ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b + p ) 2 c ( 8 s - 4 v ) ( c ( v - 2 s ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b + p ) - h - k + 2 n ( ( b + ( p + 2 c ( 2 s - v ) z ) ) 2 c ( 2 s - v ) ) 1 2 ( - h - k - 1 ) ( - b + p + 2 c ( 2 s - v ) z ) 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 + p ) ( - b + p + 2 c ( 2 s - v ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b + ( p + 2 c ( 2 s - v ) z ) ) 2 c ( 8 s - 4 v ) ) + 2 c ( 2 s - v ) ( b + ( p + 2 c ( 2 s - v ) z ) ) 2 c ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , ( b + ( p + 2 c ( 2 s - v ) z ) ) 2 c ( 8 s - 4 v ) ) ) + ( b + p ) 2 c ( 8 s - 4 v ) ( ( c ( 2 s - v ) ) 2 n ( - b p c ( 2 s - v ) k = 0 n h = 0 k ( - 1 ) k - h 4 k ( - b + p ) - h - k + 2 n ( - b + p + 2 c ( v - 2 s ) z ) h + k ( - ( - b + p + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) ) 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]]]]] ( ( - b + p ) ( - b + p + 2 c ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , ( - b + p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( v - 2 s ) - ( - b + p + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , ( - b + p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) ) + k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b + p ) - h - k + 2 n ( b + p + 2 c ( v - 2 s ) z ) h + k ( - ( b + p + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) ) 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]]]]] ( ( b + p ) ( b + p + 2 c ( v - 2 s ) z ) Γ ( 1 2 ( h + k + 1 ) , ( b + p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( v - 2 s ) - ( b + p + 2 c ( v - 2 s ) z ) 2 c ( v - 2 s ) Γ ( 1 2 ( h + k + 2 ) , ( b + p + 2 c ( v - 2 s ) z ) 2 c ( 8 s - 4 v ) ) ) ) + ( b - p ) 2 4 c s - 2 c v ( c ( v - 2 s ) ) 2 n k = 0 n h = 0 k ( - 1 ) k - h 4 k ( b + p ) - h - k + 2 n ( b + p + 2 c ( 2 s - v ) z ) h + k ( - ( b + p + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) ) 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]]]]] ( ( b + p ) ( b + p + 2 c ( 2 s - v ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( b + p + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) + 2 c ( 2 s - v ) - ( b + p + 2 c ( 2 s - v ) z ) 2 c ( 2 s - v ) Γ ( 1 2 ( h + k + 2 ) , - ( b + p + 2 c ( 2 s - v ) z ) 2 c ( 8 s - 4 v ) ) ) ) ) ) /; v + n Condition z z n p z 1 2 b z 1 2 c z v 2 -1 v -2 4 Binomial v v 2 -1 Gamma 2 n 1 -1 b -1 p z 1 2 -1 b -1 p -2 n 1 b -1 p -2 n 1 Gamma 2 n 1 b -1 p z 1 2 \$CellContext`v 2 -1 4 -1 n s 0 v -1 2 -1 c 2 s -1 v -2 n c v -1 2 s -2 n 1 Binomial v s b p 2 c 8 s -1 4 v -1 c v -1 2 s 2 n h 0 k k 0 n -1 k -1 h 4 k -1 b p -1 h -1 k 2 n b p 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 -1 h -1 k -1 -1 b p 2 c 2 s -1 v z 1 2 h k Binomial k h Binomial n k -1 b p -1 b p 2 c 2 s -1 v z 1 2 Gamma 1 2 h k 1 b p 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 2 c 2 s -1 v b p 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 Gamma 1 2 h k 2 b p 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 b p 2 c 8 s -1 4 v -1 c 2 s -1 v 2 n -1 b p c 2 s -1 v -1 h 0 k k 0 n -1 k -1 h 4 k -1 b p -1 h -1 k 2 n -1 b p 2 c v -1 2 s z 1 2 h k -1 -1 b p 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k -1 b p -1 b p 2 c v -1 2 s z 1 2 Gamma 1 2 h k 1 -1 b p 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 2 c v -1 2 s -1 -1 b p 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 Gamma 1 2 h k 2 -1 b p 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 h 0 k k 0 n -1 k -1 h 4 k b p -1 h -1 k 2 n b p 2 c v -1 2 s z 1 2 h k -1 b p 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b p b p 2 c v -1 2 s z 1 2 Gamma 1 2 h k 1 b p 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 2 c v -1 2 s -1 b p 2 c v -1 2 s z 1 2 2 c v -1 2 s -1 1 2 Gamma 1 2 h k 2 b p 2 c v -1 2 s z 1 2 2 c 8 s -1 4 v -1 b -1 p 2 4 c s -1 2 c v -1 c v -1 2 s 2 n h 0 k k 0 n -1 k -1 h 4 k b p -1 h -1 k 2 n b p 2 c 2 s -1 v z 1 2 h k -1 b p 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b p b p 2 c 2 s -1 v z 1 2 Gamma 1 2 h k 1 -1 b p 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 2 c 2 s -1 v -1 b p 2 c 2 s -1 v z 1 2 2 c 2 s -1 v -1 1 2 Gamma 1 2 h k 2 -1 b p 2 c 2 s -1 v z 1 2 2 c 8 s -1 4 v -1 v SuperPlus n [/itex]

 Rule Form

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

 2002-12-18