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/01.20.21.4810.01

 Input Form

 Integrate[z^n E^(p z^2) Sinh[b z] Cosh[c z]^v, z] == (2^(-2 - v) p^(-1 - n) ((-Binomial[v, v/2]) (-1 + Mod[v, 2]) Sum[2^(-n + q) b^(n - q) (-((b - 2 p z)^2/p))^((1/2) (-1 - q)) (-b + 2 p z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, -((b - 2 p z)^2/(4 p))], {q, 0, n}] + Binomial[v, v/2] (-1 + Mod[v, 2]) Sum[2^(-n + q) (-b)^(n - q) (b + 2 p z)^(1 + q) (-((b + 2 p z)^2/p))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((b + 2 p z)^2/(4 p))], {q, 0, n}] + E^(b^2/(4 p)) p^(1/2 + n) Sum[(Binomial[v, s] (E^((b - 2 c s + c v)^2/(4 p)) Sum[2^(-n + q) p^(-(1/2) - n) (b + 2 c s - c v)^(n - q) (-((b + 2 c s - c v - 2 p z)^2/p))^((1/2) (-1 - q)) (-b - 2 c s + c v + 2 p z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, -((b + 2 c s - c v - 2 p z)^2/(4 p))], {q, 0, n}] + E^((b + 2 c s - c v)^2/(4 p)) Sum[2^(-n + q) p^(-(1/2) - n) (b - 2 c s + c v)^(n - q) (-((b - 2 c s + c v - 2 p z)^2/p))^((1/2) (-1 - q)) (-b + 2 c s - c v + 2 p z)^(1 + q) Binomial[n, q] Gamma[(1 + q)/2, -((b - 2 c s + c v - 2 p z)^2/(4 p))], {q, 0, n}] - E^((b - 2 c s + c v)^2/(4 p)) Sum[2^(-n + q) p^(-(1/2) - n) (-b - 2 c s + c v)^(n - q) (b + 2 c s - c v + 2 p z)^(1 + q) (-((b + 2 c s - c v + 2 p z)^2/ p))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((b + 2 c s - c v + 2 p z)^2/(4 p))], {q, 0, n}] - E^((b + 2 c s - c v)^2/(4 p)) Sum[p^(-(1/2) - n) (-(b/2) + c (s - v/2))^(n - q) (b - 2 c s + c v + 2 p z)^(1 + q) (-((b - 2 c s + c v + 2 p z)^2/p))^(-(1/2) - q/2) Binomial[n, q] Gamma[(1 + q)/2, -((b - 2 c s + c v + 2 p z)^2/(4 p))], {q, 0, n}]))/E^((b^2 + c^2 (-2 s + v)^2)/(2 p)), {s, 0, Floor[(1/2) (-1 + v)]}]))/E^(b^2/(4 p)) /; Element[v, Integers] && v > 0 && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n p z 2 sinh ( b z ) cosh v ( c z ) z 2 - v - 2 - b 2 4 p p - n - 1 ( b 2 4 p ( 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 p - n - 1 2 ( 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 ) - q 2 - 1 2 ( 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 p - n - 1 2 ( - b - 2 c s + c v ) 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 p - n - 1 2 ( b - 2 c s + c v ) 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 p - n - 1 2 ( b + 2 c s - c v ) 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 ) ) ) p n + 1 2 + ( 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 ) q + 1 ( - ( b + 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 p z ) 2 4 p ) - ( 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 ) ( 2 p z - b ) 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 ) ) /; v + n Condition z z n p z 2 b z c z v 2 -1 v -2 -1 b 2 4 p -1 p -1 n -1 b 2 4 p -1 s 0 v -1 2 -1 -1 b 2 c 2 v -1 2 s 2 2 p -1 Binomial v s -1 b 2 c s -1 c v 2 4 p -1 q 0 n p -1 n -1 1 2 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 q 2 -1 -1 1 2 Binomial n q Gamma q 1 2 -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 2 q -1 n p -1 n -1 1 2 -1 b -1 2 c s c v n -1 q b 2 c s -1 c v 2 p z q 1 -1 b 2 c s -1 c v 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 b 2 c s -1 c v 2 p z 2 4 p -1 b 2 c s -1 c v 2 4 p -1 q 0 n 2 q -1 n p -1 n -1 1 2 b -1 2 c s c v 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 b -1 2 c s c v 2 4 p -1 q 0 n 2 q -1 n p -1 n -1 1 2 b 2 c s -1 c v n -1 q -1 b 2 c s -1 c v -1 2 p z 2 p -1 1 2 -1 q -1 -1 b -1 2 c s c v 2 p z q 1 Binomial n q Gamma q 1 2 -1 -1 b 2 c s -1 c v -1 2 p z 2 4 p -1 p n 1 2 Binomial v v 2 -1 \$CellContext`v 2 -1 q 0 n 2 q -1 n -1 b n -1 q b 2 p z q 1 -1 b 2 p z 2 p -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 b 2 p z 2 4 p -1 -1 Binomial v v 2 -1 \$CellContext`v 2 -1 q 0 n 2 q -1 n b n -1 q -1 b -1 2 p z 2 p -1 1 2 -1 q -1 2 p z -1 b q 1 Binomial n q Gamma q 1 2 -1 -1 b -1 2 p z 2 4 p -1 v SuperPlus n [/itex]

 Rule Form

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

 2002-12-18