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/01.20.21.1155.01

 Input Form

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

 Rule Form

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

 2002-12-18