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/01.20.21.1385.01

 Input Form

 Integrate[z^n E^(p Sqrt[z]) Sin[b z]^m Cosh[c z], z] == (-I)^m 2^(-2 - m - 2 n) (((-I^m) Binomial[m, m/2] (-1 + Mod[m, 2]) (E^(p^2/(2 c)) Sum[(-1)^(-q + r) 4^r p^(2 n - q - r) (p - 2 c Sqrt[z])^(q + r) ((p - 2 c Sqrt[z])^2/c)^ ((1/2) (-1 - q - r)) Binomial[n, r] Binomial[r, q] (p (p - 2 c Sqrt[z]) Gamma[(1/2) (1 + q + r), (p - 2 c Sqrt[z])^2/ (4 c)] - 2 c Sqrt[(p - 2 c Sqrt[z])^2/c] Gamma[(1/2) (2 + q + r), (p - 2 c Sqrt[z])^2/(4 c)]), {r, 0, n}, {q, 0, r}] + Sum[(-1)^(-q + r) 4^r p^(2 n - q - r) (p + 2 c Sqrt[z])^(q + r) (-((p + 2 c Sqrt[z])^2/c))^ ((1/2) (-1 - q - r)) Binomial[n, r] Binomial[r, q] (p (p + 2 c Sqrt[z]) Gamma[(1/2) (1 + q + r), -((p + 2 c Sqrt[z])^2/(4 c))] + 2 c Sqrt[-((p + 2 c Sqrt[z])^2/c)] Gamma[(1/2) (2 + q + r), -((p + 2 c Sqrt[z])^2/(4 c))]), {r, 0, n}, {q, 0, r}]))/ (c^(2 (1 + n)) E^(p^2/(4 c))) + Sum[(-1)^k Binomial[m, k] ((E^(p^2/(4 c + 8 I b k - 4 I b m)) Sum[(-1)^(-q + r) 4^r p^(2 n - q - r) (p - 2 (c + 2 I b k - I b m) Sqrt[z])^(q + r) ((p - 2 (c + 2 I b k - I b m) Sqrt[z])^2/(c + I b (2 k - m)))^ ((1/2) (-1 - q - r)) Binomial[n, r] Binomial[r, q] (p (p - 2 (c + 2 I b k - I b m) Sqrt[z]) Gamma[(1/2) (1 + q + r), (p - 2 (c + 2 I b k - I b m) Sqrt[z])^2/(4 (c + I b (2 k - m)))] - 2 (c + I b (2 k - m)) Sqrt[(p - 2 (c + 2 I b k - I b m) Sqrt[z])^2/(c + I b (2 k - m))] Gamma[(1/2) (2 + q + r), (p - 2 (c + 2 I b k - I b m) Sqrt[z])^2/(4 (c + I b (2 k - m)))]), {r, 0, n}, {q, 0, r}])/ (-c - 2 I b k + I b m)^(2 (1 + n)) + (E^(p^2/(-4 c - 8 I b k + 4 I b m) + I m Pi) Sum[(-1)^(-q + r) 4^r p^(2 n - q - r) (p + 2 (c + 2 I b k - I b m) Sqrt[z])^(q + r) (-((p + 2 (c + 2 I b k - I b m) Sqrt[z])^2/(c + 2 I b k - I b m)))^ ((1/2) (-1 - q - r)) Binomial[n, r] Binomial[r, q] (p (p + 2 (c + 2 I b k - I b m) Sqrt[z]) Gamma[(1/2) (1 + q + r), -((p + 2 (c + 2 I b k - I b m) Sqrt[z])^2/(4 (c + 2 I b k - I b m)))] + 2 (c + 2 I b k - I b m) Sqrt[-((p + 2 (c + 2 I b k - I b m) Sqrt[z])^2/(c + 2 I b k - I b m))] Gamma[(1/2) (2 + q + r), -((p + 2 (c + 2 I b k - I b m) Sqrt[z])^2/(4 (c + 2 I b k - I b m)))]), {r, 0, n}, {q, 0, r}])/(c + 2 I b k - I b m)^ (2 (1 + n)) + (E^(p^2/(4 c - 8 I b k + 4 I b m) + I m Pi) Sum[(-1)^(-q + r) 4^r p^(2 n - q - r) (p - 2 (c - 2 I b k + I b m) Sqrt[z])^(q + r) ((p - 2 (c - 2 I b k + I b m) Sqrt[z])^2/(c - 2 I b k + I b m))^ ((1/2) (-1 - q - r)) Binomial[n, r] Binomial[r, q] (p (p - 2 (c - 2 I b k + I b m) Sqrt[z]) Gamma[(1/2) (1 + q + r), (p - 2 (c - 2 I b k + I b m) Sqrt[z])^2/(4 (c - 2 I b k + I b m))] - 2 (c - 2 I b k + I b m) Sqrt[(p - 2 (c - 2 I b k + I b m) Sqrt[z])^2/(c - 2 I b k + I b m)] Gamma[(1/2) (2 + q + r), (p - 2 (c - 2 I b k + I b m) Sqrt[z])^2/(4 (c - 2 I b k + I b m))]), {r, 0, n}, {q, 0, r}])/(-c + 2 I b k - I b m)^(2 (1 + n)) + (E^(p^2/(-4 c + 8 I b k - 4 I b m)) Sum[(-1)^(-q + r) 4^r p^(2 n - q - r) (p + 2 (c - 2 I b k + I b m) Sqrt[z])^(q + r) (-((p + 2 (c - 2 I b k + I b m) Sqrt[z])^2/(c - 2 I b k + I b m)))^ ((1/2) (-1 - q - r)) Binomial[n, r] Binomial[r, q] (p (p + 2 (c - 2 I b k + I b m) Sqrt[z]) Gamma[(1/2) (1 + q + r), -((p + 2 (c - 2 I b k + I b m) Sqrt[z])^2/(4 (c - 2 I b k + I b m)))] + 2 (c - 2 I b k + I b m) Sqrt[-((p + 2 (c - 2 I b k + I b m) Sqrt[z])^2/(c - 2 I b k + I b m))] Gamma[(1/2) (2 + q + r), -((p + 2 (c - 2 I b k + I b m) Sqrt[z])^2/(4 (c - 2 I b k + I b m)))]), {r, 0, n}, {q, 0, r}])/(c - 2 I b k + I b m)^ (2 (1 + n))), {k, 0, Floor[(1/2) (-1 + m)]}]) /; Element[m, Integers] && m > 0 && Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n p z sin m ( b z ) cosh ( c z ) z ( - ) m 2 - m - 2 n - 2 ( - m c - 2 ( n + 1 ) - p 2 4 c ( 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 ) ( p 2 2 c r = 0 n q = 0 r ( - 1 ) r - q 4 r p 2 n - q - r ( p - 2 c z ) q + r ( ( p - 2 c z ) 2 c ) 1 2 ( - q - r - 1 ) ( n r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["r", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( r q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["r", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( p - 2 c z ) Γ ( 1 2 ( q + r + 1 ) , ( p - 2 c z ) 2 4 c ) - 2 c ( p - 2 c z ) 2 c Γ ( 1 2 ( q + r + 2 ) , ( p - 2 c z ) 2 4 c ) ) + r = 0 n q = 0 r ( - 1 ) r - q 4 r p 2 n - q - r ( 2 z c + p ) q + r ( - ( 2 z c + p ) 2 c ) 1 2 ( - q - r - 1 ) ( n r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["r", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( r q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["r", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( 2 z c + p ) Γ ( 1 2 ( q + r + 1 ) , - ( 2 z c + p ) 2 4 c ) + 2 - ( 2 z c + p ) 2 c c Γ ( 1 2 ( q + r + 2 ) , - ( 2 z c + p ) 2 4 c ) ) ) + k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p 2 4 c + 8 b k - 4 b m ( r = 0 n q = 0 r ( - 1 ) r - q 4 r p 2 n - q - r ( p - 2 ( c + 2 b k - b m ) z ) q + r ( ( p - 2 ( c + 2 b k - b m ) z ) 2 c + b ( 2 k - m ) ) 1 2 ( - q - r - 1 ) ( n r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["r", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( r q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["r", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( p - 2 ( c + 2 b k - b m ) z ) Γ ( 1 2 ( q + r + 1 ) , ( p - 2 ( c + 2 b k - b m ) z ) 2 4 ( c + b ( 2 k - m ) ) ) - 2 ( c + b ( 2 k - m ) ) ( p - 2 ( c + 2 b k - b m ) z ) 2 c + b ( 2 k - m ) Γ ( 1 2 ( q + r + 2 ) , ( p - 2 ( c + 2 b k - b m ) z ) 2 4 ( c + b ( 2 k - m ) ) ) ) ) ( - c - 2 b k + b m ) - 2 ( n + 1 ) + p 2 4 c - 8 b k + 4 b m + m π ( - c + 2 b k - b m ) - 2 ( n + 1 ) r = 0 n q = 0 r ( - 1 ) r - q 4 r p 2 n - q - r ( p - 2 ( c - 2 b k + b m ) z ) q + r ( ( p - 2 ( c - 2 b k + b m ) z ) 2 c - 2 b k + b m ) 1 2 ( - q - r - 1 ) ( n r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["r", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( r q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["r", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( p - 2 ( c - 2 b k + b m ) z ) Γ ( 1 2 ( q + r + 1 ) , ( p - 2 ( c - 2 b k + b m ) z ) 2 4 ( c - 2 b k + b m ) ) - 2 ( c - 2 b k + b m ) ( p - 2 ( c - 2 b k + b m ) z ) 2 c - 2 b k + b m Γ ( 1 2 ( q + r + 2 ) , ( p - 2 ( c - 2 b k + b m ) z ) 2 4 ( c - 2 b k + b m ) ) ) + p 2 - 4 c + 8 b k - 4 b m ( c - 2 b k + b m ) - 2 ( n + 1 ) r = 0 n q = 0 r ( - 1 ) r - q 4 r p 2 n - q - r ( 2 z ( c - 2 b k + b m ) + p ) q + r ( - ( 2 z ( c - 2 b k + b m ) + p ) 2 c - 2 b k + b m ) 1 2 ( - q - r - 1 ) ( n r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["r", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( r q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["r", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( 2 z ( c - 2 b k + b m ) + p ) Γ ( 1 2 ( q + r + 1 ) , - ( 2 z ( c - 2 b k + b m ) + p ) 2 4 ( c - 2 b k + b m ) ) + 2 - ( 2 z ( c - 2 b k + b m ) + p ) 2 c - 2 b k + b m ( c - 2 b k + b m ) Γ ( 1 2 ( q + r + 2 ) , - ( 2 z ( c - 2 b k + b m ) + p ) 2 4 ( c - 2 b k + b m ) ) ) + p 2 - 4 c - 8 b k + 4 b m + m π ( c + 2 b k - b m ) - 2 ( n + 1 ) r = 0 n q = 0 r ( - 1 ) r - q 4 r p 2 n - q - r ( 2 z ( c + 2 b k - b m ) + p ) q + r ( - ( 2 z ( c + 2 b k - b m ) + p ) 2 c + 2 b k - b m ) 1 2 ( - q - r - 1 ) ( n r ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["r", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( r q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["r", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p ( 2 z ( c + 2 b k - b m ) + p ) Γ ( 1 2 ( q + r + 1 ) , - ( 2 z ( c + 2 b k - b m ) + p ) 2 4 ( c + 2 b k - b m ) ) + 2 - ( 2 z ( c + 2 b k - b m ) + p ) 2 c + 2 b k - b m ( c + 2 b k - b m ) Γ ( 1 2 ( q + r + 2 ) , - ( 2 z ( c + 2 b k - b m ) + p ) 2 4 ( c + 2 b k - b m ) ) ) ) ) /; m + n Condition z z n p z 1 2 b z m c z -1 m 2 -1 m -1 2 n -2 -1 m c -2 n 1 -1 p 2 4 c -1 Binomial m m 2 -1 \$CellContext`m 2 -1 p 2 2 c -1 q 0 r r 0 n -1 r -1 q 4 r p 2 n -1 q -1 r p -1 2 c z 1 2 q r p -1 2 c z 1 2 2 c -1 1 2 -1 q -1 r -1 Binomial n r Binomial r q p p -1 2 c z 1 2 Gamma 1 2 q r 1 p -1 2 c z 1 2 2 4 c -1 -1 2 c p -1 2 c z 1 2 2 c -1 1 2 Gamma 1 2 q r 2 p -1 2 c z 1 2 2 4 c -1 q 0 r r 0 n -1 r -1 q 4 r p 2 n -1 q -1 r 2 z 1 2 c p q r -1 2 z 1 2 c p 2 c -1 1 2 -1 q -1 r -1 Binomial n r Binomial r q p 2 z 1 2 c p Gamma 1 2 q r 1 -1 2 z 1 2 c p 2 4 c -1 2 -1 2 z 1 2 c p 2 c -1 1 2 c Gamma 1 2 q r 2 -1 2 z 1 2 c p 2 4 c -1 k 0 m -1 2 -1 -1 k Binomial m k p 2 4 c 8 b k -1 4 b m -1 q 0 r r 0 n -1 r -1 q 4 r p 2 n -1 q -1 r p -1 2 c 2 b k -1 b m z 1 2 q r p -1 2 c 2 b k -1 b m z 1 2 2 c b 2 k -1 m -1 1 2 -1 q -1 r -1 Binomial n r Binomial r q p p -1 2 c 2 b k -1 b m z 1 2 Gamma 1 2 q r 1 p -1 2 c 2 b k -1 b m z 1 2 2 4 c b 2 k -1 m -1 -1 2 c b 2 k -1 m p -1 2 c 2 b k -1 b m z 1 2 2 c b 2 k -1 m -1 1 2 Gamma 1 2 q r 2 p -1 2 c 2 b k -1 b m z 1 2 2 4 c b 2 k -1 m -1 -1 c -1 2 b k b m -2 n 1 p 2 4 c -1 8 b k 4 b m -1 m -1 c 2 b k -1 b m -2 n 1 q 0 r r 0 n -1 r -1 q 4 r p 2 n -1 q -1 r p -1 2 c -1 2 b k b m z 1 2 q r p -1 2 c -1 2 b k b m z 1 2 2 c -1 2 b k b m -1 1 2 -1 q -1 r -1 Binomial n r Binomial r q p p -1 2 c -1 2 b k b m z 1 2 Gamma 1 2 q r 1 p -1 2 c -1 2 b k b m z 1 2 2 4 c -1 2 b k b m -1 -1 2 c -1 2 b k b m p -1 2 c -1 2 b k b m z 1 2 2 c -1 2 b k b m -1 1 2 Gamma 1 2 q r 2 p -1 2 c -1 2 b k b m z 1 2 2 4 c -1 2 b k b m -1 p 2 -4 c 8 b k -1 4 b m -1 c -1 2 b k b m -2 n 1 q 0 r r 0 n -1 r -1 q 4 r p 2 n -1 q -1 r 2 z 1 2 c -1 2 b k b m p q r -1 2 z 1 2 c -1 2 b k b m p 2 c -1 2 b k b m -1 1 2 -1 q -1 r -1 Binomial n r Binomial r q p 2 z 1 2 c -1 2 b k b m p Gamma 1 2 q r 1 -1 2 z 1 2 c -1 2 b k b m p 2 4 c -1 2 b k b m -1 2 -1 2 z 1 2 c -1 2 b k b m p 2 c -1 2 b k b m -1 1 2 c -1 2 b k b m Gamma 1 2 q r 2 -1 2 z 1 2 c -1 2 b k b m p 2 4 c -1 2 b k b m -1 p 2 -4 c -1 8 b k 4 b m -1 m c 2 b k -1 b m -2 n 1 q 0 r r 0 n -1 r -1 q 4 r p 2 n -1 q -1 r 2 z 1 2 c 2 b k -1 b m p q r -1 2 z 1 2 c 2 b k -1 b m p 2 c 2 b k -1 b m -1 1 2 -1 q -1 r -1 Binomial n r Binomial r q p 2 z 1 2 c 2 b k -1 b m p Gamma 1 2 q r 1 -1 2 z 1 2 c 2 b k -1 b m p 2 4 c 2 b k -1 b m -1 2 -1 2 z 1 2 c 2 b k -1 b m p 2 c 2 b k -1 b m -1 1 2 c 2 b k -1 b m Gamma 1 2 q r 2 -1 2 z 1 2 c 2 b k -1 b m p 2 4 c 2 b k -1 b m -1 m SuperPlus n [/itex]

 Rule Form

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

 2002-12-18