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/01.20.21.1043.01

 Input Form

 Integrate[z^n Cos[b Sqrt[z] + d z + e] Cosh[c z], z] == 2^(-3 - 2 n) ((E^(b^2/(4 (-c - I d)) - I e) Sum[(-1)^(-h + j) 4^j ((-I) b)^(-h - j + 2 n) ((-I) b + 2 (-c - I d) Sqrt[z])^(h + j) (-(((-I) b + 2 (-c - I d) Sqrt[z])^2/(-c - I d)))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((-I) b ((-I) b + 2 (-c - I d) Sqrt[z]) Gamma[(1/2) (1 + h + j), -(((-I) b + 2 (-c - I d) Sqrt[z])^2/(4 (-c - I d)))] + 2 (-c - I d) Sqrt[-(((-I) b + 2 (-c - I d) Sqrt[z])^2/(-c - I d))] Gamma[(1/2) (2 + h + j), -(((-I) b + 2 (-c - I d) Sqrt[z])^2/ (4 (-c - I d)))]), {j, 0, n}, {h, 0, j}])/ (c + I d)^(2 (1 + n)) + (E^(b^2/(4 (c - I d)) - I e) Sum[(-1)^(-h + j) 4^j ((-I) b)^(-h - j + 2 n) ((-I) b + 2 (c - I d) Sqrt[z])^(h + j) (-(((-I) b + 2 (c - I d) Sqrt[z])^2/(c - I d)))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] ((-I) b ((-I) b + 2 (c - I d) Sqrt[z]) Gamma[(1/2) (1 + h + j), -(((-I) b + 2 (c - I d) Sqrt[z])^2/ (4 (c - I d)))] + 2 (c - I d) Sqrt[-(((-I) b + 2 (c - I d) Sqrt[z])^2/(c - I d))] Gamma[(1/2) (2 + h + j), -(((-I) b + 2 (c - I d) Sqrt[z])^2/ (4 (c - I d)))]), {j, 0, n}, {h, 0, j}])/(c - I d)^(2 (1 + n)) + (E^(b^2/(4 (-c + I d)) + I e) Sum[(-1)^(-h + j) 4^j (I b)^(-h - j + 2 n) (I b + 2 (-c + I d) Sqrt[z])^(h + j) (-((I b + 2 (-c + I d) Sqrt[z])^2/(-c + I d)))^((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (I b (I b + 2 (-c + I d) Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I b + 2 (-c + I d) Sqrt[z])^2/ (4 (-c + I d)))] + 2 (-c + I d) Sqrt[-((I b + 2 (-c + I d) Sqrt[z])^2/(-c + I d))] Gamma[(1/2) (2 + h + j), -((I b + 2 (-c + I d) Sqrt[z])^2/ (4 (-c + I d)))]), {j, 0, n}, {h, 0, j}])/ (-c + I d)^(2 (1 + n)) + (E^(b^2/(4 (c + I d)) + I e) Sum[(-1)^(-h + j) 4^j (I b)^(-h - j + 2 n) (I b + 2 (c + I d) Sqrt[z])^ (h + j) (-((I b + 2 (c + I d) Sqrt[z])^2/(c + I d)))^ ((1/2) (-1 - h - j)) Binomial[j, h] Binomial[n, j] (I b (I b + 2 (c + I d) Sqrt[z]) Gamma[(1/2) (1 + h + j), -((I b + 2 (c + I d) Sqrt[z])^2/(4 (c + I d)))] + 2 (c + I d) Sqrt[-((I b + 2 (c + I d) Sqrt[z])^2/(c + I d))] Gamma[(1/2) (2 + h + j), -((I b + 2 (c + I d) Sqrt[z])^2/ (4 (c + I d)))]), {j, 0, n}, {h, 0, j}])/ (c + I d)^(2 (1 + n))) /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n cos ( z b + e + d z ) cosh ( c z ) z 2 - 2 n - 3 ( b 2 4 ( d - c ) + e ( j = 0 n h = 0 j ( - 1 ) j - h 4 j ( b ) - h - j + 2 n ( 2 z ( d - c ) + b ) h + j ( - ( 2 z ( d - c ) + b ) 2 d - c ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b ( 2 z ( d - c ) + b ) Γ ( 1 2 ( h + j + 1 ) , - ( 2 z ( d - c ) + b ) 2 4 ( d - c ) ) + 2 - ( 2 z ( d - c ) + b ) 2 d - c ( d - c ) Γ ( 1 2 ( h + j + 2 ) , - ( 2 z ( d - c ) + b ) 2 4 ( d - c ) ) ) ) ( d - c ) - 2 ( n + 1 ) + ( c + d ) - 2 ( n + 1 ) b 2 4 ( c + d ) + e j = 0 n h = 0 j ( - 1 ) j - h 4 j ( b ) - h - j + 2 n ( 2 z ( c + d ) + b ) h + j ( - ( 2 z ( c + d ) + b ) 2 c + d ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b ( 2 z ( c + d ) + b ) Γ ( 1 2 ( h + j + 1 ) , - ( 2 z ( c + d ) + b ) 2 4 ( c + d ) ) + 2 - ( 2 z ( c + d ) + b ) 2 c + d ( c + d ) Γ ( 1 2 ( h + j + 2 ) , - ( 2 z ( c + d ) + b ) 2 4 ( c + d ) ) ) + ( c + d ) - 2 ( n + 1 ) b 2 4 ( - c - d ) - e j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - b ) - h - j + 2 n ( 2 ( - c - d ) z - b ) h + j ( - ( 2 ( - c - d ) z - b ) 2 - c - d ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 ( - c - d ) - ( 2 ( - c - d ) z - b ) 2 - c - d Γ ( 1 2 ( h + j + 2 ) , - ( 2 ( - c - d ) z - b ) 2 4 ( - c - d ) ) - b ( 2 ( - c - d ) z - b ) Γ ( 1 2 ( h + j + 1 ) , - ( 2 ( - c - d ) z - b ) 2 4 ( - c - d ) ) ) + ( c - d ) - 2 ( n + 1 ) b 2 4 ( c - d ) - e j = 0 n h = 0 j ( - 1 ) j - h 4 j ( - b ) - h - j + 2 n ( 2 ( c - d ) z - b ) h + j ( - ( 2 ( c - d ) z - b ) 2 c - d ) 1 2 ( - h - j - 1 ) ( j h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["j", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["j", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 ( c - d ) - ( 2 ( c - d ) z - b ) 2 c - d Γ ( 1 2 ( h + j + 2 ) , - ( 2 ( c - d ) z - b ) 2 4 ( c - d ) ) - b ( 2 ( c - d ) z - b ) Γ ( 1 2 ( h + j + 1 ) , - ( 2 ( c - d ) z - b ) 2 4 ( c - d ) ) ) ) /; n Condition z z n z 1 2 b e d z c z 2 -2 n -3 b 2 4 d -1 c -1 e h 0 j j 0 n -1 j -1 h 4 j b -1 h -1 j 2 n 2 z 1 2 d -1 c b h j -1 2 z 1 2 d -1 c b 2 d -1 c -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j b 2 z 1 2 d -1 c b Gamma 1 2 h j 1 -1 2 z 1 2 d -1 c b 2 4 d -1 c -1 2 -1 2 z 1 2 d -1 c b 2 d -1 c -1 1 2 d -1 c Gamma 1 2 h j 2 -1 2 z 1 2 d -1 c b 2 4 d -1 c -1 d -1 c -2 n 1 c d -2 n 1 b 2 4 c d -1 e h 0 j j 0 n -1 j -1 h 4 j b -1 h -1 j 2 n 2 z 1 2 c d b h j -1 2 z 1 2 c d b 2 c d -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j b 2 z 1 2 c d b Gamma 1 2 h j 1 -1 2 z 1 2 c d b 2 4 c d -1 2 -1 2 z 1 2 c d b 2 c d -1 1 2 c d Gamma 1 2 h j 2 -1 2 z 1 2 c d b 2 4 c d -1 c d -2 n 1 b 2 4 -1 c -1 d -1 -1 e h 0 j j 0 n -1 j -1 h 4 j -1 b -1 h -1 j 2 n 2 -1 c -1 d z 1 2 -1 b h j -1 2 -1 c -1 d z 1 2 -1 b 2 -1 c -1 d -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j 2 -1 c -1 d -1 2 -1 c -1 d z 1 2 -1 b 2 -1 c -1 d -1 1 2 Gamma 1 2 h j 2 -1 2 -1 c -1 d z 1 2 -1 b 2 4 -1 c -1 d -1 -1 b 2 -1 c -1 d z 1 2 -1 b Gamma 1 2 h j 1 -1 2 -1 c -1 d z 1 2 -1 b 2 4 -1 c -1 d -1 c -1 d -2 n 1 b 2 4 c -1 d -1 -1 e h 0 j j 0 n -1 j -1 h 4 j -1 b -1 h -1 j 2 n 2 c -1 d z 1 2 -1 b h j -1 2 c -1 d z 1 2 -1 b 2 c -1 d -1 1 2 -1 h -1 j -1 Binomial j h Binomial n j 2 c -1 d -1 2 c -1 d z 1 2 -1 b 2 c -1 d -1 1 2 Gamma 1 2 h j 2 -1 2 c -1 d z 1 2 -1 b 2 4 c -1 d -1 -1 b 2 c -1 d z 1 2 -1 b Gamma 1 2 h j 1 -1 2 c -1 d z 1 2 -1 b 2 4 c -1 d -1 n [/itex]

 Rule Form

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

 2002-12-18