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/01.20.21.4812.01

 Input Form

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

 Rule Form

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

 2002-12-18