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/01.20.21.1116.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18