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/01.20.21.3644.01

 Input Form

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

 Standard Form

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 MathML Form

 z n cos ( b z 2 + d z + e ) cosh v ( c z 2 + f z + g ) z - 2 - v - 2 ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["v", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - v mod 2 \$CellContext`v 2 ) ( d 2 4 b - e ( j = 0 n 2 j - n ( d ) n - j ( - d - 2 b z ) j + 1 ( - ( - d - 2 b z ) 2 b ) 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 - 2 b z ) 2 4 b ) ) ( - b ) - n - 1 + ( b ) - n - 1 e - d 2 4 b j = 0 n 2 j - n ( - d ) n - j ( d + 2 b z ) j + 1 ( ( d + 2 b z ) 2 b ) 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 + 2 b z ) 2 4 b ) ) - 2 - v - 2 s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( - ( - d + f ( 2 s - v ) ) 2 4 ( - b + c ( 2 s - v ) ) - e + g ( 2 s - v ) ( j = 0 n 2 j - n ( d - f ( 2 s - v ) ) n - j ( - d + f ( 2 s - v ) + 2 ( - b + c ( 2 s - v ) ) z ) j + 1 ( - ( - d + f ( 2 s - v ) + 2 ( - b + c ( 2 s - v ) ) z ) 2 - b + c ( 2 s - v ) ) 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 + f ( 2 s - v ) + 2 ( - b + c ( 2 s - v ) ) z ) 2 4 ( - b + c ( 2 s - v ) ) ) ) ( - b + c ( 2 s - v ) ) - n - 1 + - ( d + f ( 2 s - v ) ) 2 4 ( b + c ( 2 s - v ) ) + e + g ( 2 s - v ) ( b + c ( 2 s - v ) ) - n - 1 j = 0 n 2 j - n ( - d - f ( 2 s - v ) ) n - j ( d + f ( 2 s - v ) + 2 ( b + c ( 2 s - v ) ) z ) j + 1 ( - ( d + f ( 2 s - v ) + 2 ( b + c ( 2 s - v ) ) z ) 2 b + c ( 2 s - v ) ) 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 + f ( 2 s - v ) + 2 ( b + c ( 2 s - v ) ) z ) 2 4 ( b + c ( 2 s - v ) ) ) + - ( - d + f ( v - 2 s ) ) 2 4 ( - b + c ( v - 2 s ) ) - e + g ( v - 2 s ) ( - b + c ( v - 2 s ) ) - n - 1 j = 0 n 2 j - n ( d - f ( v - 2 s ) ) n - j ( - d + f ( v - 2 s ) + 2 ( - b + c ( v - 2 s ) ) z ) j + 1 ( - ( - d + f ( v - 2 s ) + 2 ( - b + c ( v - 2 s ) ) z ) 2 - b + c ( v - 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 + f ( v - 2 s ) + 2 ( - b + c ( v - 2 s ) ) z ) 2 4 ( - b + c ( v - 2 s ) ) ) + - ( d + f ( v - 2 s ) ) 2 4 ( b + c ( v - 2 s ) ) + e + g ( v - 2 s ) ( b + c ( v - 2 s ) ) - n - 1 j = 0 n 2 j - n ( - d - f ( v - 2 s ) ) n - j ( d + f ( v - 2 s ) + 2 ( b + c ( v - 2 s ) ) z ) j + 1 ( - ( d + f ( v - 2 s ) + 2 ( b + c ( v - 2 s ) ) z ) 2 b + c ( v - 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 + f ( v - 2 s ) + 2 ( b + c ( v - 2 s ) ) z ) 2 4 ( b + c ( v - 2 s ) ) ) ) /; n v + Condition z z n b z 2 d z e c z 2 f z g v -1 2 -1 v -2 Binomial v v 2 -1 1 -1 \$CellContext`v 2 d 2 4 b -1 -1 e j 0 n 2 j -1 n d n -1 j -1 d -1 2 b z j 1 -1 -1 d -1 2 b z 2 b -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 -1 d -1 2 b z 2 4 b -1 -1 b -1 n -1 b -1 n -1 e -1 d 2 4 b -1 j 0 n 2 j -1 n -1 d n -1 j d 2 b z j 1 d 2 b z 2 b -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 d 2 b z 2 4 b -1 -1 2 -1 v -2 s 0 v -1 2 -1 Binomial v s -1 -1 d f 2 s -1 v 2 4 -1 b c 2 s -1 v -1 -1 e g 2 s -1 v j 0 n 2 j -1 n d -1 f 2 s -1 v n -1 j -1 d f 2 s -1 v 2 -1 b c 2 s -1 v z j 1 -1 -1 d f 2 s -1 v 2 -1 b c 2 s -1 v z 2 -1 b c 2 s -1 v -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 -1 d f 2 s -1 v 2 -1 b c 2 s -1 v z 2 4 -1 b c 2 s -1 v -1 -1 b c 2 s -1 v -1 n -1 -1 d f 2 s -1 v 2 4 b c 2 s -1 v -1 e g 2 s -1 v b c 2 s -1 v -1 n -1 j 0 n 2 j -1 n -1 d -1 f 2 s -1 v n -1 j d f 2 s -1 v 2 b c 2 s -1 v z j 1 -1 d f 2 s -1 v 2 b c 2 s -1 v z 2 b c 2 s -1 v -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 d f 2 s -1 v 2 b c 2 s -1 v z 2 4 b c 2 s -1 v -1 -1 -1 d f v -1 2 s 2 4 -1 b c v -1 2 s -1 -1 e g v -1 2 s -1 b c v -1 2 s -1 n -1 j 0 n 2 j -1 n d -1 f v -1 2 s n -1 j -1 d f v -1 2 s 2 -1 b c v -1 2 s z j 1 -1 -1 d f v -1 2 s 2 -1 b c v -1 2 s z 2 -1 b c v -1 2 s -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 -1 d f v -1 2 s 2 -1 b c v -1 2 s z 2 4 -1 b c v -1 2 s -1 -1 d f v -1 2 s 2 4 b c v -1 2 s -1 e g v -1 2 s b c v -1 2 s -1 n -1 j 0 n 2 j -1 n -1 d -1 f v -1 2 s n -1 j d f v -1 2 s 2 b c v -1 2 s z j 1 -1 d f v -1 2 s 2 b c v -1 2 s z 2 b c v -1 2 s -1 1 2 -1 j -1 Binomial n j Gamma j 1 2 -1 -1 d f v -1 2 s 2 b c v -1 2 s z 2 4 b c v -1 2 s -1 n v SuperPlus [/itex]

 Rule Form

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

 2002-12-18