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/01.20.21.1440.01

 Input Form

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

 Standard Form

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 MathML Form

 z n p z cos m ( b z ) cosh ( c z 2 ) z 2 - m - 2 c - n - 1 ( - p 2 4 c ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox[FractionBox["m", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( m mod 2 \$CellContext`m 2 - 1 ) ( q = 0 n 2 q - n ( - p ) n - q ( p + 2 c z ) q + 1 ( - ( p + 2 c z ) 2 c ) 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 z ) 2 4 c ) - ( - 1 ) n p 2 2 c q = 0 n 2 q - n ( - p ) n - q ( p - 2 c z ) q + 1 ( ( p - 2 c z ) 2 c ) 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 z ) 2 4 c ) ) - s = 0 m - 1 2 ( p + b ( m - 2 s ) ) 2 4 c ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - ( - 1 ) n q = 0 n 2 q - n ( - p - b ( m - 2 s ) ) n - q ( p + b ( m - 2 s ) - 2 c z ) q + 1 ( ( p + b ( m - 2 s ) - 2 c z ) 2 c ) 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 + b ( m - 2 s ) - 2 c z ) 2 4 c ) - ( - 1 ) n - b p ( m - 2 s ) c q = 0 n 2 q - n ( b ( m - 2 s ) - p ) n - q ( p - b ( m - 2 s ) - 2 c z ) q + 1 ( ( p - b ( m - 2 s ) - 2 c z ) 2 c ) 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 - b ( m - 2 s ) - 2 c z ) 2 4 c ) + - ( p + b ( m - 2 s ) ) 2 2 c q = 0 n 2 q - n ( - p - b ( m - 2 s ) ) n - q ( p + b ( m - 2 s ) + 2 c z ) q + 1 ( - ( p + b ( m - 2 s ) + 2 c z ) 2 c ) 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 + b ( m - 2 s ) + 2 c z ) 2 4 c ) + b 2 ( m - 2 s ) 2 - p 2 2 c q = 0 n 2 q - n ( b ( m - 2 s ) - p ) n - q ( p - b ( m - 2 s ) + 2 c z ) q + 1 ( ( b ( m - 2 s ) + ( p + 2 c z ) ) 2 c ) 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 - b ( m - 2 s ) + 2 c z ) 2 4 c ) ) ) /; m + n Condition z z n p z b z m c z 2 2 -1 m -2 c -1 n -1 -1 p 2 4 c -1 Binomial m m 2 -1 \$CellContext`m 2 -1 q 0 n 2 q -1 n -1 p n -1 q p 2 c z q 1 -1 p 2 c z 2 c -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 p 2 c z 2 4 c -1 -1 -1 n p 2 2 c -1 q 0 n 2 q -1 n -1 p n -1 q p -1 2 c z q 1 p -1 2 c z 2 c -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p -1 2 c z 2 4 c -1 -1 s 0 m -1 2 -1 p b m -1 2 s 2 4 c -1 Binomial m s -1 -1 n q 0 n 2 q -1 n -1 p -1 b m -1 2 s n -1 q p b m -1 2 s -1 2 c z q 1 p b m -1 2 s -1 2 c z 2 c -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p b m -1 2 s -1 2 c z 2 4 c -1 -1 -1 n -1 b p m -1 2 s c -1 q 0 n 2 q -1 n b m -1 2 s -1 p n -1 q p -1 b m -1 2 s -1 2 c z q 1 p -1 b m -1 2 s -1 2 c z 2 c -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 p -1 b m -1 2 s -1 2 c z 2 4 c -1 -1 p b m -1 2 s 2 2 c -1 q 0 n 2 q -1 n -1 p -1 b m -1 2 s n -1 q p b m -1 2 s 2 c z q 1 -1 p b m -1 2 s 2 c z 2 c -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 p b m -1 2 s 2 c z 2 4 c -1 b 2 m -1 2 s 2 -1 p 2 2 c -1 q 0 n 2 q -1 n b m -1 2 s -1 p n -1 q p -1 b m -1 2 s 2 c z q 1 b m -1 2 s p 2 c z 2 c -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 p -1 b m -1 2 s 2 c z 2 4 c -1 m SuperPlus n [/itex]

 Rule Form

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

 2002-12-18