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/01.20.21.3974.01

 Input Form

 Integrate[z^n E^(p z^2) Cos[b z] Cosh[c z]^v, z] == 2^(-2 - v) (E^(b^2/(4 p)) p^(-1 - n) Binomial[v, v/2] (-1 + Mod[v, 2]) (Sum[2^(-n + q) ((-I) b)^(n - q) ((b - 2 I p z)^2/p)^((1/2) (-1 - q)) (I b + 2 p z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (b - 2 I p z)^2/(4 p)], {q, 0, n}] + Sum[2^(-n + q) (I b)^(n - q) ((b + 2 I p z)^2/p)^((1/2) (-1 - q)) ((-I) b + 2 p z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, (b + 2 I p z)^2/(4 p)], {q, 0, n}]) - p^(-1 - n) Sum[E^((b^2 - c^2 (-2 s + v)^2)/(2 p)) Binomial[v, s] (E^((I b - 2 c s + c v)^2/(4 p)) Sum[2^(-n + q) ((-I) b + c (-2 s + v))^(n - q) (-(((-I) b - 2 c s + c v - 2 p z)^2/p))^((1/2) (-1 - q)) (I b + 2 c s - c v + 2 p z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, -(((-I) b - 2 c s + c v - 2 p z)^2/(4 p))], {q, 0, n}] + E^(((-I) b - 2 c s + c v)^2/(4 p)) Sum[2^(-n + q) (I b + c (-2 s + v))^(n - q) (-((I b - 2 c s + c v - 2 p z)^2/p))^((1/2) (-1 - q)) ((-I) b + 2 c s - c v + 2 p z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, -((I b - 2 c s + c v - 2 p z)^2/(4 p))], {q, 0, n}] + E^((I b - 2 c s + c v)^2/(4 p)) Sum[((I b)/2 + c s - (c v)/2)^(n - q) ((-I) b - 2 c s + c v + 2 p z)^ (1 + q) (-(((-I) b - 2 c s + c v + 2 p z)^2/p))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -(((-I) b - 2 c s + c v + 2 p z)^2/ (4 p))], {q, 0, n}] + E^(((-I) b - 2 c s + c v)^2/(4 p)) Sum[(-((I b)/2) + c (s - v/2))^(n - q) (I b - 2 c s + c v + 2 p z)^ (1 + q) (-((I b - 2 c s + c v + 2 p z)^2/p))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((I b - 2 c s + c v + 2 p z)^2/ (4 p))], {q, 0, n}]), {s, 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 2 cos ( b z ) cosh v ( c z ) z 2 - v - 2 ( b 2 4 p p - n - 1 ( 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 ) ( q = 0 n 2 q - n ( b ) n - q ( ( b + 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( - b + 2 p z ) q + 1 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( b + 2 p z ) 2 4 p ) + q = 0 n 2 q - n ( - b ) n - q ( ( b - 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( b + 2 p z ) q + 1 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , ( b - 2 p z ) 2 4 p ) ) - p - n - 1 s = 0 v - 1 2 b 2 - c 2 ( v - 2 s ) 2 2 p ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( b - 2 c s + c v ) 2 4 p q = 0 n ( b 2 + c s - c v 2 ) n - q ( - b - 2 c s + c v + 2 p z ) q + 1 ( - ( - b - 2 c s + c v + 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( - b - 2 c s + c v + 2 p z ) 2 4 p ) + ( - b - 2 c s + c v ) 2 4 p q = 0 n ( c ( s - v 2 ) - b 2 ) n - q ( b - 2 c s + c v + 2 p z ) q + 1 ( - ( b - 2 c s + c v + 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( b - 2 c s + c v + 2 p z ) 2 4 p ) + ( b - 2 c s + c v ) 2 4 p q = 0 n 2 q - n ( - b + c ( v - 2 s ) ) n - q ( - ( - b - 2 c s + c v - 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( b + 2 c s - c v + 2 p z ) q + 1 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( - b - 2 c s + c v - 2 p z ) 2 4 p ) + ( - b - 2 c s + c v ) 2 4 p q = 0 n 2 q - n ( b + c ( v - 2 s ) ) n - q ( - ( b - 2 c s + c v - 2 p z ) 2 p ) 1 2 ( - q - 1 ) ( - b + 2 c s - c v + 2 p z ) q + 1 ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] Γ ( q + 1 2 , - ( b - 2 c s + c v - 2 p z ) 2 4 p ) ) ) /; v TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] v > 0 n Condition z z n p z 2 b z c z v 2 -1 v -2 b 2 4 p -1 p -1 n -1 Binomial v v 2 -1 \$CellContext`v 2 -1 q 0 n 2 q -1 n b n -1 q b 2 p z 2 p -1 1 2 -1 q -1 -1 b 2 p z q 1 Binomial n q Gamma q 1 2 -1 b 2 p z 2 4 p -1 q 0 n 2 q -1 n -1 b n -1 q b -1 2 p z 2 p -1 1 2 -1 q -1 b 2 p z q 1 Binomial n q Gamma q 1 2 -1 b -1 2 p z 2 4 p -1 -1 p -1 n -1 s 0 v -1 2 -1 b 2 -1 c 2 v -1 2 s 2 2 p -1 Binomial v s b -1 2 c s c v 2 4 p -1 q 0 n b 2 -1 c s -1 c v 2 -1 n -1 q -1 b -1 2 c s c v 2 p z q 1 -1 -1 b -1 2 c s c v 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 -1 b -1 2 c s c v 2 p z 2 4 p -1 -1 b -1 2 c s c v 2 4 p -1 q 0 n c s -1 v 2 -1 -1 b 2 -1 n -1 q b -1 2 c s c v 2 p z q 1 -1 b -1 2 c s c v 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 b -1 2 c s c v 2 p z 2 4 p -1 b -1 2 c s c v 2 4 p -1 q 0 n 2 q -1 n -1 b c v -1 2 s n -1 q -1 -1 b -1 2 c s c v -1 2 p z 2 p -1 1 2 -1 q -1 b 2 c s -1 c v 2 p z q 1 Binomial n q Gamma q 1 2 -1 -1 -1 b -1 2 c s c v -1 2 p z 2 4 p -1 -1 b -1 2 c s c v 2 4 p -1 q 0 n 2 q -1 n b c v -1 2 s n -1 q -1 b -1 2 c s c v -1 2 p z 2 p -1 1 2 -1 q -1 -1 b 2 c s -1 c v 2 p z q 1 Binomial n q Gamma q 1 2 -1 -1 b -1 2 c s c v -1 2 p z 2 4 p -1 v v 0 n [/itex]

 Rule Form

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

 2002-12-18