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/01.20.21.3972.01

 Input Form

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