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/01.20.21.1388.01

 Input Form

 Integrate[z^n E^(p z) Sin[b z]^m Cosh[c z^2], z] == (2^(-2 - m) (4 E^(p^2/(2 c)) Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[2^(-2 - n + q) (-c)^(-1 - n) (-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}] + 4 Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[2^(-2 - n + q) c^(-1 - n) (-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[((-1)^k 2^(-1 - n) c^(-1 - n) Binomial[m, k] ((-c^n) E^((8 I b k p + 3 p^2 + 4 I c m Pi)/(4 c)) Sum[(I b (-2 k + m) - p)^(n - q) (2 I b k - I b m + p - 2 c z)^ (1 + q) Binomial[n, q] ExpIntegralE[(1 - q)/2, (2 I b k - I b m + p - 2 c z)^2/(4 c)], {q, 0, n}] - 2^n c^n E^((p (4 I b m + 3 p))/(4 c)) Sum[((I b (k - m/2) - p/2)^(n - q) (I b (-2 k + m) + p - 2 c z)^( 1 + q) Binomial[n, q] ExpIntegralE[(1 - q)/2, (I b (-2 k + m) + p - 2 c z)^2/(4 c)])/2^q, {q, 0, n}] + (-c)^n E^((2 b^2 (-2 k + m)^2 + p^2)/(4 c)) (E^((I m (b p + c Pi))/c) Sum[(I b (-2 k + m) - p)^(n - q) (2 I b k - I b m + p + 2 c z)^(1 + q) Binomial[n, q] ExpIntegralE[(1 - q)/2, -((2 I b k - I b m + p + 2 c z)^2/ (4 c))], {q, 0, n}] + 2^n E^((2 I b k p)/c) Sum[((I b (k - m/2) - p/2)^(n - q) (I b (-2 k + m) + p + 2 c z)^ (1 + q) Binomial[n, q] ExpIntegralE[(1 - q)/2, -((I b (-2 k + m) + p + 2 c z)^2/(4 c))])/2^q, {q, 0, n}])))/ ((-c)^n E^((b^2 (-2 k + m)^2 + 2 I b (2 k + m) p + 2 (p^2 + I c m Pi))/(4 c))), {k, 0, Floor[(1/2) (-1 + m)]}]))/ E^(p^2/(4 c)) /; Element[m, Integers] && m > 0 && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n p z sin m ( b z ) cosh ( c z 2 ) z 2 - m - 2 - p 2 4 c ( 4 p 2 2 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 - n + q - 2 ( - c ) - n - 1 ( - 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 ) + 4 ( 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 - n + q - 2 c - n - 1 ( - 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 ) - p 2 4 c k = 0 m - 1 2 ( - 1 ) k 2 - n - 1 ( - c ) - n c - n - 1 - b 2 ( m - 2 k ) 2 + 2 b ( 2 k + m ) p + 2 ( p 2 + π c m ) 4 c ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 b 2 ( m - 2 k ) 2 + p 2 4 c ( m ( π c + b p ) c q = 0 n ( b ( m - 2 k ) - p ) n - q ( 2 b k - b m + p + 2 c 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]]]]] E TagBox["E", ExpIntegralE] 1 - q 2 ( - ( 2 b k - b m + p + 2 c z ) 2 4 c ) + 2 n 2 b k p c q = 0 n 2 - q ( b ( k - m 2 ) - p 2 ) n - q ( b ( m - 2 k ) + p + 2 c 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]]]]] E TagBox["E", ExpIntegralE] 1 - q 2 ( - ( b ( m - 2 k ) + p + 2 c z ) 2 4 c ) ) ( - c ) n - c n 3 p 2 + 8 b k p + 4 π c m 4 c q = 0 n ( b ( m - 2 k ) - p ) n - q ( 2 b k - b m + p - 2 c 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]]]]] E TagBox["E", ExpIntegralE] 1 - q 2 ( ( 2 b k - b m + p - 2 c z ) 2 4 c ) - 2 n c n p ( 4 b m + 3 p ) 4 c q = 0 n 2 - q ( b ( k - m 2 ) - p 2 ) n - q ( b ( m - 2 k ) + p - 2 c 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]]]]] E TagBox["E", ExpIntegralE] 1 - q 2 ( ( b ( m - 2 k ) + p - 2 c z ) 2 4 c ) ) ) /; m + n Condition z z n p z b z m c z 2 2 -1 m -2 -1 p 2 4 c -1 4 p 2 2 c -1 Binomial m m 2 -1 \$CellContext`m 2 -1 q 0 n 2 -1 n q -2 -1 c -1 n -1 -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 4 Binomial m m 2 -1 \$CellContext`m 2 -1 q 0 n 2 -1 n q -2 c -1 n -1 -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 p 2 4 c -1 k 0 m -1 2 -1 -1 k 2 -1 n -1 -1 c -1 n c -1 n -1 -1 b 2 m -1 2 k 2 2 b 2 k m p 2 p 2 c m 4 c -1 Binomial m k 2 b 2 m -1 2 k 2 p 2 4 c -1 m c b p c -1 q 0 n b m -1 2 k -1 p n -1 q 2 b k -1 b m p 2 c z q 1 Binomial n q ExpIntegralE 1 -1 q 2 -1 -1 2 b k -1 b m p 2 c z 2 4 c -1 2 n 2 b k p c -1 q 0 n 2 -1 q b k -1 m 2 -1 -1 p 2 -1 n -1 q b m -1 2 k p 2 c z q 1 Binomial n q ExpIntegralE 1 -1 q 2 -1 -1 b m -1 2 k p 2 c z 2 4 c -1 -1 c n -1 c n 3 p 2 8 b k p 4 c m 4 c -1 q 0 n b m -1 2 k -1 p n -1 q 2 b k -1 b m p -1 2 c z q 1 Binomial n q ExpIntegralE 1 -1 q 2 -1 2 b k -1 b m p -1 2 c z 2 4 c -1 -1 2 n c n p 4 b m 3 p 4 c -1 q 0 n 2 -1 q b k -1 m 2 -1 -1 p 2 -1 n -1 q b m -1 2 k p -1 2 c z q 1 Binomial n q ExpIntegralE 1 -1 q 2 -1 b m -1 2 k p -1 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