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/01.20.21.1409.01

 Input Form

 Integrate[z^n E^(p Sqrt[z]) Cos[b z] Cosh[c z], z] == 2^(-3 - 2 n) ((E^(p^2/(4 c - 4 I b)) Sum[(-1)^(-q + r) 4^r p^(2 n - q - r) (p - 2 (c - I b) Sqrt[z])^(q + r) ((p - 2 (c - I b) Sqrt[z])^2/ (c - I b))^((1/2) (-1 - q - r)) Binomial[n, r] Binomial[r, q] (p (p - 2 (c - I b) Sqrt[z]) Gamma[(1/2) (1 + q + r), (p - 2 (c - I b) Sqrt[z])^2/(4 (c - I b))] - 2 (c - I b) Sqrt[(p - 2 (c - I b) Sqrt[z])^2/(c - I b)] Gamma[(1/2) (2 + q + r), (p - 2 (c - I b) Sqrt[z])^2/ (4 (c - I b))]), {r, 0, n}, {q, 0, r}])/(-c + I b)^(2 (1 + n)) + (E^(p^2/(-4 c + 4 I b)) Sum[(-1)^(-q + r) 4^r p^(2 n - q - r) (p + 2 (c - I b) Sqrt[z])^(q + r) (-((p + 2 (c - I b) Sqrt[z])^2/(c - I b)))^((1/2) (-1 - q - r)) Binomial[n, r] Binomial[r, q] (p (p + 2 (c - I b) Sqrt[z]) Gamma[(1/2) (1 + q + r), -((p + 2 (c - I b) Sqrt[z])^2/ (4 (c - I b)))] + 2 (c - I b) Sqrt[-((p + 2 (c - I b) Sqrt[z])^2/ (c - I b))] Gamma[(1/2) (2 + q + r), -((p + 2 (c - I b) Sqrt[z])^2/(4 (c - I b)))]), {r, 0, n}, {q, 0, r}])/(c - I b)^(2 (1 + n)) + (E^(p^2/(4 c + 4 I b)) Sum[(-1)^(-q + r) 4^r p^(2 n - q - r) (p - 2 (c + I b) Sqrt[z])^(q + r) ((p - 2 (c + I b) Sqrt[z])^2/ (c + I b))^((1/2) (-1 - q - r)) Binomial[n, r] Binomial[r, q] (p (p - 2 (c + I b) Sqrt[z]) Gamma[(1/2) (1 + q + r), (p - 2 (c + I b) Sqrt[z])^2/(4 (c + I b))] - 2 (c + I b) Sqrt[(p - 2 (c + I b) Sqrt[z])^2/(c + I b)] Gamma[(1/2) (2 + q + r), (p - 2 (c + I b) Sqrt[z])^2/ (4 (c + I b))]), {r, 0, n}, {q, 0, r}])/(-c - I b)^(2 (1 + n)) + (E^(p^2/(-4 c - 4 I b)) Sum[(-1)^(-q + r) 4^r p^(2 n - q - r) (p + 2 (c + I b) Sqrt[z])^(q + r) (-((p + 2 (c + I b) Sqrt[z])^2/(c + I b)))^((1/2) (-1 - q - r)) Binomial[n, r] Binomial[r, q] (p (p + 2 (c + I b) Sqrt[z]) Gamma[(1/2) (1 + q + r), -((p + 2 (c + I b) Sqrt[z])^2/ (4 (c + I b)))] + 2 (c + I b) Sqrt[-((p + 2 (c + I b) Sqrt[z])^2/ (c + I b))] Gamma[(1/2) (2 + q + r), -((p + 2 (c + I b) Sqrt[z])^2/(4 (c + I b)))]), {r, 0, n}, {q, 0, r}])/(c + I b)^(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 ( p 2 4 c - 4 b ( r = 0 n q = 0 r ( - 1 ) r - q 4 r p 2 n - q - r ( p - 2 ( c - b ) z ) q + r ( ( p - 2 ( c - b ) z ) 2 c - b ) 1 2 ( - q - r - 1 ) ( n r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["r", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( r q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["r", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( p - 2 ( c - b ) z ) Γ ( 1 2 ( q + r + 1 ) , ( p - 2 ( c - b ) z ) 2 4 ( c - b ) ) - 2 ( c - b ) ( p - 2 ( c - b ) z ) 2 c - b Γ ( 1 2 ( q + r + 2 ) , ( p - 2 ( c - b ) z ) 2 4 ( c - b ) ) ) ) ( b - c ) - 2 ( n + 1 ) + ( - c - b ) - 2 ( n + 1 ) p 2 4 c + 4 b r = 0 n q = 0 r ( - 1 ) r - q 4 r p 2 n - q - r ( p - 2 ( c + b ) z ) q + r ( ( p - 2 ( c + b ) z ) 2 c + b ) 1 2 ( - q - r - 1 ) ( n r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["r", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( r q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["r", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( p - 2 ( c + b ) z ) Γ ( 1 2 ( q + r + 1 ) , ( p - 2 ( c + b ) z ) 2 4 ( c + b ) ) - 2 ( c + b ) ( p - 2 ( c + b ) z ) 2 c + b Γ ( 1 2 ( q + r + 2 ) , ( p - 2 ( c + b ) z ) 2 4 ( c + b ) ) ) + ( c + b ) - 2 ( n + 1 ) p 2 - 4 c - 4 b r = 0 n q = 0 r ( - 1 ) r - q 4 r p 2 n - q - r ( 2 z ( c + b ) + p ) q + r ( - ( 2 z ( c + b ) + p ) 2 c + b ) 1 2 ( - q - r - 1 ) ( n r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["r", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( r q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["r", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( 2 z ( c + b ) + p ) Γ ( 1 2 ( q + r + 1 ) , - ( 2 z ( c + b ) + p ) 2 4 ( c + b ) ) + 2 - ( 2 z ( c + b ) + p ) 2 c + b ( c + b ) Γ ( 1 2 ( q + r + 2 ) , - ( 2 z ( c + b ) + p ) 2 4 ( c + b ) ) ) + ( c - b ) - 2 ( n + 1 ) p 2 4 b - 4 c r = 0 n q = 0 r ( - 1 ) r - q 4 r p 2 n - q - r ( 2 z ( c - b ) + p ) q + r ( - ( 2 z ( c - b ) + p ) 2 c - b ) 1 2 ( - q - r - 1 ) ( n r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["r", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( r q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["r", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( 2 z ( c - b ) + p ) Γ ( 1 2 ( q + r + 1 ) , - ( 2 z ( c - b ) + p ) 2 4 ( c - b ) ) + 2 - ( 2 z ( c - b ) + p ) 2 c - b ( c - b ) Γ ( 1 2 ( q + r + 2 ) , - ( 2 z ( c - b ) + p ) 2 4 ( c - b ) ) ) ) /; n Condition z z n p z 1 2 b z c z 2 -2 n -3 p 2 4 c -1 4 b -1 q 0 r r 0 n -1 r -1 q 4 r p 2 n -1 q -1 r p -1 2 c -1 b z 1 2 q r p -1 2 c -1 b z 1 2 2 c -1 b -1 1 2 -1 q -1 r -1 Binomial n r Binomial r q p p -1 2 c -1 b z 1 2 Gamma 1 2 q r 1 p -1 2 c -1 b z 1 2 2 4 c -1 b -1 -1 2 c -1 b p -1 2 c -1 b z 1 2 2 c -1 b -1 1 2 Gamma 1 2 q r 2 p -1 2 c -1 b z 1 2 2 4 c -1 b -1 b -1 c -2 n 1 -1 c -1 b -2 n 1 p 2 4 c 4 b -1 q 0 r r 0 n -1 r -1 q 4 r p 2 n -1 q -1 r p -1 2 c b z 1 2 q r p -1 2 c b z 1 2 2 c b -1 1 2 -1 q -1 r -1 Binomial n r Binomial r q p p -1 2 c b z 1 2 Gamma 1 2 q r 1 p -1 2 c b z 1 2 2 4 c b -1 -1 2 c b p -1 2 c b z 1 2 2 c b -1 1 2 Gamma 1 2 q r 2 p -1 2 c b z 1 2 2 4 c b -1 c b -2 n 1 p 2 -4 c -1 4 b -1 q 0 r r 0 n -1 r -1 q 4 r p 2 n -1 q -1 r 2 z 1 2 c b p q r -1 2 z 1 2 c b p 2 c b -1 1 2 -1 q -1 r -1 Binomial n r Binomial r q p 2 z 1 2 c b p Gamma 1 2 q r 1 -1 2 z 1 2 c b p 2 4 c b -1 2 -1 2 z 1 2 c b p 2 c b -1 1 2 c b Gamma 1 2 q r 2 -1 2 z 1 2 c b p 2 4 c b -1 c -1 b -2 n 1 p 2 4 b -1 4 c -1 q 0 r r 0 n -1 r -1 q 4 r p 2 n -1 q -1 r 2 z 1 2 c -1 b p q r -1 2 z 1 2 c -1 b p 2 c -1 b -1 1 2 -1 q -1 r -1 Binomial n r Binomial r q p 2 z 1 2 c -1 b p Gamma 1 2 q r 1 -1 2 z 1 2 c -1 b p 2 4 c -1 b -1 2 -1 2 z 1 2 c -1 b p 2 c -1 b -1 1 2 c -1 b Gamma 1 2 q r 2 -1 2 z 1 2 c -1 b p 2 4 c -1 b -1 n [/itex]

 Rule Form

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

 2002-12-18