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/01.20.21.1441.01

 Input Form

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

 Standard Form

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 MathML Form

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

 Rule Form

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

 2002-12-18