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/01.20.21.1138.01

 Input Form

 Integrate[z^n Cos[b z^2 + d z]^m Cosh[c z^2], z] == (-2^(-2 - m)) z^(1 + n) Binomial[m, m/2] (((-c) z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, (-c) z^2] + (c z^2)^((1/2) (-1 - n)) Gamma[(1 + n)/2, c z^2]) (1 - Mod[m, 2]) - 2^(-2 - m) Sum[Binomial[m, s] (((-c - I b (m - 2 s))^(-1 - n) Sum[2^(j - n) (I d (m - 2 s))^(-j + n) ((-I) d (m - 2 s) + 2 (-c - I b (m - 2 s)) z)^(1 + j) (-(((-I) d (m - 2 s) + 2 (-c - I b (m - 2 s)) z)^2/ (-c - I b (m - 2 s))))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -(((-I) d (m - 2 s) + 2 (-c - I b (m - 2 s)) z)^2/ (4 (-c - I b (m - 2 s))))], {j, 0, n}])/ E^((I d^2 (m - 2 s)^2)/(4 (I c - b m + 2 b s))) + ((c - I b (m - 2 s))^(-1 - n) Sum[2^(j - n) (I d (m - 2 s))^(-j + n) ((-I) d (m - 2 s) + 2 (c - I b (m - 2 s)) z)^(1 + j) (-(((-I) d (m - 2 s) + 2 (c - I b (m - 2 s)) z)^2/ (c - I b (m - 2 s))))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -(((-I) d (m - 2 s) + 2 (c - I b (m - 2 s)) z)^2/ (4 (c - I b (m - 2 s))))], {j, 0, n}])/ E^((I d^2 (m - 2 s)^2)/(4 ((-I) c - b m + 2 b s))) + E^((I d^2 (m - 2 s)^2)/(4 ((-I) c - b m + 2 b s))) (-c + I b (m - 2 s))^(-1 - n) Sum[2^(j - n) ((-I) d (m - 2 s))^ (-j + n) (I d (m - 2 s) + 2 (-c + I b (m - 2 s)) z)^(1 + j) (-((I d (m - 2 s) + 2 (-c + I b (m - 2 s)) z)^2/ (-c + I b (m - 2 s))))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((I d (m - 2 s) + 2 (-c + I b (m - 2 s)) z)^2/ (4 (-c + I b (m - 2 s))))], {j, 0, n}] + E^((I d^2 (m - 2 s)^2)/(4 (I c - b m + 2 b s))) (c + I b (m - 2 s))^ (-1 - n) Sum[2^(j - n) ((-I) d (m - 2 s))^(-j + n) (I d (m - 2 s) + 2 (c + I b (m - 2 s)) z)^(1 + j) (-((I d (m - 2 s) + 2 (c + I b (m - 2 s)) z)^2/ (c + I b (m - 2 s))))^((1/2) (-1 - j)) Binomial[n, j] Gamma[(1 + j)/2, -((I d (m - 2 s) + 2 (c + I b (m - 2 s)) z)^2/ (4 (c + I 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 2 ) z - 2 - m - 2 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( Γ ( n + 1 2 , - c z 2 ) ( - c z 2 ) 1 2 ( - n - 1 ) + ( c z 2 ) 1 2 ( - n - 1 ) Γ ( n + 1 2 , c z 2 ) ) ( 1 - m mod 2 \$CellContext`m 2 ) z n + 1 - 2 - m - 2 s = 0 m - 1 2 ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( d 2 ( m - 2 s ) 2 4 ( - c - b m + 2 b s ) ( j = 0 n 2 j - n ( - d ( m - 2 s ) ) n - j ( d ( m - 2 s ) + 2 ( b ( m - 2 s ) - c ) z ) j + 1 ( - ( d ( m - 2 s ) + 2 ( b ( m - 2 s ) - c ) z ) 2 b ( m - 2 s ) - c ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( j + 1 2 , - ( d ( m - 2 s ) + 2 ( b ( m - 2 s ) - c ) z ) 2 4 ( b ( m - 2 s ) - c ) ) ) ( b ( m - 2 s ) - c ) - n - 1 + d 2 ( m - 2 s ) 2 4 ( c - b m + 2 b s ) ( c + b ( m - 2 s ) ) - n - 1 j = 0 n 2 j - n ( - d ( m - 2 s ) ) n - j ( d ( m - 2 s ) + 2 ( c + b ( m - 2 s ) ) z ) j + 1 ( - ( d ( m - 2 s ) + 2 ( c + b ( m - 2 s ) ) z ) 2 c + b ( m - 2 s ) ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( j + 1 2 , - ( d ( m - 2 s ) + 2 ( c + b ( m - 2 s ) ) z ) 2 4 ( c + b ( m - 2 s ) ) ) + - d 2 ( m - 2 s ) 2 4 ( c - b m + 2 b s ) ( - c - b ( m - 2 s ) ) - n - 1 j = 0 n 2 j - n ( d ( m - 2 s ) ) n - j ( 2 ( - c - b ( m - 2 s ) ) z - d ( m - 2 s ) ) j + 1 ( - ( 2 ( - c - b ( m - 2 s ) ) z - d ( m - 2 s ) ) 2 - c - b ( m - 2 s ) ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( j + 1 2 , - ( 2 ( - c - b ( m - 2 s ) ) z - d ( m - 2 s ) ) 2 4 ( - c - b ( m - 2 s ) ) ) + - d 2 ( m - 2 s ) 2 4 ( - c - b m + 2 b s ) ( c - b ( m - 2 s ) ) - n - 1 j = 0 n 2 j - n ( d ( m - 2 s ) ) n - j ( 2 ( c - b ( m - 2 s ) ) z - d ( m - 2 s ) ) j + 1 ( - ( 2 ( c - b ( m - 2 s ) ) z - d ( m - 2 s ) ) 2 c - b ( m - 2 s ) ) 1 2 ( - j - 1 ) ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( j + 1 2 , - ( 2 ( c - b ( m - 2 s ) ) z - d ( m - 2 s ) ) 2 4 ( c - b ( m - 2 s ) ) ) ) /; n m + Condition z z n b z 2 d z m c z 2 -1 2 -1 m -2 Binomial m m 2 -1 Gamma n 1 2 -1 -1 c z 2 -1 c z 2 1 2 -1 n -1 c z 2 1 2 -1 n -1 Gamma n 1 2 -1 c z 2 1 -1 \$CellContext`m 2 z n 1 -1 2 -1 m -2 s 0 m -1 2 -1 Binomial m s d 2 m -1 2 s 2 4 -1 c -1 b m 2 b s -1 j 0 n 2 j -1 n -1 d m -1 2 s n -1 j d m -1 2 s 2 b m -1 2 s -1 c z j 1 -1 d m -1 2 s 2 b m -1 2 s -1 c z 2 b m -1 2 s -1 c -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 d m -1 2 s 2 b m -1 2 s -1 c z 2 4 b m -1 2 s -1 c -1 b m -1 2 s -1 c -1 n -1 d 2 m -1 2 s 2 4 c -1 b m 2 b s -1 c b m -1 2 s -1 n -1 j 0 n 2 j -1 n -1 d m -1 2 s n -1 j d m -1 2 s 2 c b m -1 2 s z j 1 -1 d m -1 2 s 2 c b m -1 2 s z 2 c b m -1 2 s -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 d m -1 2 s 2 c b m -1 2 s z 2 4 c b m -1 2 s -1 -1 d 2 m -1 2 s 2 4 c -1 b m 2 b s -1 -1 c -1 b m -1 2 s -1 n -1 j 0 n 2 j -1 n d m -1 2 s n -1 j 2 -1 c -1 b m -1 2 s z -1 d m -1 2 s j 1 -1 2 -1 c -1 b m -1 2 s z -1 d m -1 2 s 2 -1 c -1 b m -1 2 s -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 2 -1 c -1 b m -1 2 s z -1 d m -1 2 s 2 4 -1 c -1 b m -1 2 s -1 -1 d 2 m -1 2 s 2 4 -1 c -1 b m 2 b s -1 c -1 b m -1 2 s -1 n -1 j 0 n 2 j -1 n d m -1 2 s n -1 j 2 c -1 b m -1 2 s z -1 d m -1 2 s j 1 -1 2 c -1 b m -1 2 s z -1 d m -1 2 s 2 c -1 b m -1 2 s -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 2 c -1 b m -1 2 s z -1 d m -1 2 s 2 4 c -1 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