html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cosh

 http://functions.wolfram.com/01.20.21.1413.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18