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/01.20.21.1437.01

 Input Form

 Integrate[z^n E^(p Sqrt[z]) Cos[b z]^m Cosh[c z], z] == (2^(-2 - m - 2 n) ((-E^(p^2/(2 c))) Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p - 2 c Sqrt[z])^(h + k) ((p - 2 c Sqrt[z])^2/c)^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p - 2 c Sqrt[z]) Gamma[(1/2) (1 + h + k), (p - 2 c Sqrt[z])^2/(4 c)] - 2 c Sqrt[(p - 2 c Sqrt[z])^2/c] Gamma[(1/2) (2 + h + k), (p - 2 c Sqrt[z])^2/(4 c)]), {k, 0, n}, {h, 0, k}] - Binomial[m, m/2] (-1 + Mod[m, 2]) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p + 2 c Sqrt[z])^(h + k) (-((p + 2 c Sqrt[z])^2/c))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p + 2 c Sqrt[z]) Gamma[(1/2) (1 + h + k), -((p + 2 c Sqrt[z])^2/(4 c))] + 2 c Sqrt[-((p + 2 c Sqrt[z])^2/c)] Gamma[(1/2) (2 + h + k), -((p + 2 c Sqrt[z])^2/(4 c))]), {k, 0, n}, {h, 0, k}] + c^(2 (1 + n)) E^(p^2/(4 c)) Sum[Binomial[m, u] ((E^(p^2/(4 c - 4 I b (m - 2 u))) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p - 2 (c - I b (m - 2 u)) Sqrt[z])^(h + k) ((p - 2 (c - I b (m - 2 u)) Sqrt[z])^2/(c - I b (m - 2 u)))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p - 2 (c - I b (m - 2 u)) Sqrt[z]) Gamma[ (1/2) (1 + h + k), (p - 2 (c - I b (m - 2 u)) Sqrt[z])^2/ (4 (c - I b (m - 2 u)))] - 2 (c - I b (m - 2 u)) Sqrt[ (p - 2 (c - I b (m - 2 u)) Sqrt[z])^2/(c - I b (m - 2 u))] Gamma[(1/2) (2 + h + k), (p - 2 (c - I b (m - 2 u)) Sqrt[z])^2/ (4 (c - I b (m - 2 u)))]), {k, 0, n}, {h, 0, k}])/ (-c + I b (m - 2 u))^(2 (1 + n)) + (E^(p^2/(-4 c + 4 I b m - 8 I b u)) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p + 2 (c - I b (m - 2 u)) Sqrt[z])^(h + k) ((p + 2 (c - I b (m - 2 u)) Sqrt[z])^2/(-c + I b (m - 2 u)))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p + 2 (c - I b (m - 2 u)) Sqrt[z]) Gamma[(1/2) (1 + h + k), (p + 2 (c - I b (m - 2 u)) Sqrt[z])^2/(-4 c + 4 I b (m - 2 u))] + 2 (c - I b (m - 2 u)) Sqrt[ (p + 2 (c - I b (m - 2 u)) Sqrt[z])^2/(-c + I b (m - 2 u))] Gamma[(1/2) (2 + h + k), (p + 2 (c - I b (m - 2 u)) Sqrt[z])^2/ (-4 c + 4 I b (m - 2 u))]), {k, 0, n}, {h, 0, k}])/ (c - I b (m - 2 u))^(2 (1 + n)) + (E^(p^2/(4 c + 4 I b (m - 2 u))) Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p - 2 (c + I b (m - 2 u)) Sqrt[z])^(h + k) ((p - 2 (c + I b (m - 2 u)) Sqrt[z])^2/(c + I b (m - 2 u)))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p - 2 (c + I b (m - 2 u)) Sqrt[z]) Gamma[ (1/2) (1 + h + k), (p - 2 (c + I b (m - 2 u)) Sqrt[z])^2/ (4 (c + I b (m - 2 u)))] - 2 (c + I b (m - 2 u)) Sqrt[ (p - 2 (c + I b (m - 2 u)) Sqrt[z])^2/(c + I b (m - 2 u))] Gamma[(1/2) (2 + h + k), (p - 2 (c + I b (m - 2 u)) Sqrt[z])^2/ (4 (c + I b (m - 2 u)))]), {k, 0, n}, {h, 0, k}])/ (-c - I b (m - 2 u))^(2 (1 + n)) + Sum[(-1)^(-h + k) 4^k p^(-h - k + 2 n) (p + 2 (c + I b (m - 2 u)) Sqrt[z])^(h + k) (-((p + 2 (c + I b (m - 2 u)) Sqrt[z])^2/(c + I b (m - 2 u))))^ ((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (p (p + 2 (c + I b (m - 2 u)) Sqrt[z]) Gamma[(1/2) (1 + h + k), -((p + 2 (c + I b (m - 2 u)) Sqrt[z])^2/(4 (c + I b (m - 2 u))))] + 2 (c + I b (m - 2 u)) Sqrt[-((p + 2 (c + I b (m - 2 u)) Sqrt[z])^2/(c + I b (m - 2 u)))] Gamma[(1/2) (2 + h + k), -((p + 2 (c + I b (m - 2 u)) Sqrt[z])^2/(4 (c + I b (m - 2 u))))]), {k, 0, n}, {h, 0, k}]/ (E^(p^2/(4 c + 4 I b (m - 2 u))) (c + I b (m - 2 u))^(2 (1 + n)))), {u, 0, Floor[(1/2) (-1 + m)]}]))/(c^(2 (1 + n)) E^(p^2/(4 c))) /; 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 n - 2 c - 2 ( n + 1 ) - p 2 4 c ( p 2 4 c c 2 ( n + 1 ) u = 0 m - 1 2 ( m u ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["u", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p 2 4 c - 4 b ( m - 2 u ) ( b ( m - 2 u ) - c ) - 2 ( n + 1 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p - 2 ( c - b ( m - 2 u ) ) z ) h + k ( ( p - 2 ( c - b ( m - 2 u ) ) z ) 2 c - b ( m - 2 u ) ) 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]]]]] ( p ( p - 2 ( c - b ( m - 2 u ) ) z ) Γ ( 1 2 ( h + k + 1 ) , ( p - 2 ( c - b ( m - 2 u ) ) z ) 2 4 ( c - b ( m - 2 u ) ) ) - 2 ( c - b ( m - 2 u ) ) ( p - 2 ( c - b ( m - 2 u ) ) z ) 2 c - b ( m - 2 u ) Γ ( 1 2 ( h + k + 2 ) , ( p - 2 ( c - b ( m - 2 u ) ) z ) 2 4 ( c - b ( m - 2 u ) ) ) ) + p 2 4 c + 4 b ( m - 2 u ) ( - c - b ( m - 2 u ) ) - 2 ( n + 1 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p - 2 ( c + b ( m - 2 u ) ) z ) h + k ( ( p - 2 ( c + b ( m - 2 u ) ) z ) 2 c + b ( m - 2 u ) ) 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]]]]] ( p ( p - 2 ( c + b ( m - 2 u ) ) z ) Γ ( 1 2 ( h + k + 1 ) , ( p - 2 ( c + b ( m - 2 u ) ) z ) 2 4 ( c + b ( m - 2 u ) ) ) - 2 ( c + b ( m - 2 u ) ) ( p - 2 ( c + b ( m - 2 u ) ) z ) 2 c + b ( m - 2 u ) Γ ( 1 2 ( h + k + 2 ) , ( p - 2 ( c + b ( m - 2 u ) ) z ) 2 4 ( c + b ( m - 2 u ) ) ) ) + - p 2 4 c + 4 b ( m - 2 u ) ( c + b ( m - 2 u ) ) - 2 ( n + 1 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p + 2 ( c + b ( m - 2 u ) ) z ) h + k ( - ( p + 2 ( c + b ( m - 2 u ) ) z ) 2 c + b ( m - 2 u ) ) 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]]]]] ( p ( p + 2 ( c + b ( m - 2 u ) ) z ) Γ ( 1 2 ( h + k + 1 ) , - ( p + 2 ( c + b ( m - 2 u ) ) z ) 2 4 ( c + b ( m - 2 u ) ) ) + 2 - ( p + 2 ( c + b ( m - 2 u ) ) z ) 2 c + b ( m - 2 u ) ( c + b ( m - 2 u ) ) Γ ( 1 2 ( h + k + 2 ) , - ( p + 2 ( c + b ( m - 2 u ) ) z ) 2 4 ( c + b ( m - 2 u ) ) ) ) + p 2 - 4 c + 4 b m - 8 b u ( c - b ( m - 2 u ) ) - 2 ( n + 1 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p + 2 ( c - b ( m - 2 u ) ) z ) h + k ( ( p + 2 ( c - b ( m - 2 u ) ) z ) 2 b ( m - 2 u ) - c ) 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]]]]] ( p ( p + 2 ( c - b ( m - 2 u ) ) z ) Γ ( 1 2 ( h + k + 1 ) , ( p + 2 ( c - b ( m - 2 u ) ) z ) 2 4 b ( m - 2 u ) - 4 c ) + 2 ( p + 2 ( c - b ( m - 2 u ) ) z ) 2 b ( m - 2 u ) - c ( c - b ( m - 2 u ) ) Γ ( 1 2 ( h + k + 2 ) , ( p + 2 ( c - b ( m - 2 u ) ) z ) 2 4 b ( m - 2 u ) - 4 c ) ) ) - p 2 2 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 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( p - 2 c z ) h + k ( ( p - 2 c z ) 2 c ) 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]]]]] ( p ( p - 2 c z ) Γ ( 1 2 ( h + k + 1 ) , ( p - 2 c z ) 2 4 c ) - 2 c ( p - 2 c z ) 2 c Γ ( 1 2 ( h + k + 2 ) , ( p - 2 c z ) 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 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k p - h - k + 2 n ( 2 z c + p ) h + k ( - ( 2 z c + p ) 2 c ) 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]]]]] ( p ( 2 z c + p ) Γ ( 1 2 ( h + k + 1 ) , - ( 2 z c + p ) 2 4 c ) + 2 - ( 2 z c + p ) 2 c c Γ ( 1 2 ( h + k + 2 ) , - ( 2 z c + p ) 2 4 c ) ) ) /; m + n Condition z z n p z 1 2 b z m c z 2 -1 m -1 2 n -2 c -2 n 1 -1 p 2 4 c -1 p 2 4 c -1 c 2 n 1 u 0 m -1 2 -1 Binomial m u p 2 4 c -1 4 b m -1 2 u -1 b m -1 2 u -1 c -2 n 1 h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p -1 2 c -1 b m -1 2 u z 1 2 h k p -1 2 c -1 b m -1 2 u z 1 2 2 c -1 b m -1 2 u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p -1 2 c -1 b m -1 2 u z 1 2 Gamma 1 2 h k 1 p -1 2 c -1 b m -1 2 u z 1 2 2 4 c -1 b m -1 2 u -1 -1 2 c -1 b m -1 2 u p -1 2 c -1 b m -1 2 u z 1 2 2 c -1 b m -1 2 u -1 1 2 Gamma 1 2 h k 2 p -1 2 c -1 b m -1 2 u z 1 2 2 4 c -1 b m -1 2 u -1 p 2 4 c 4 b m -1 2 u -1 -1 c -1 b m -1 2 u -2 n 1 h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p -1 2 c b m -1 2 u z 1 2 h k p -1 2 c b m -1 2 u z 1 2 2 c b m -1 2 u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p -1 2 c b m -1 2 u z 1 2 Gamma 1 2 h k 1 p -1 2 c b m -1 2 u z 1 2 2 4 c b m -1 2 u -1 -1 2 c b m -1 2 u p -1 2 c b m -1 2 u z 1 2 2 c b m -1 2 u -1 1 2 Gamma 1 2 h k 2 p -1 2 c b m -1 2 u z 1 2 2 4 c b m -1 2 u -1 -1 p 2 4 c 4 b m -1 2 u -1 c b m -1 2 u -2 n 1 h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p 2 c b m -1 2 u z 1 2 h k -1 p 2 c b m -1 2 u z 1 2 2 c b m -1 2 u -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p 2 c b m -1 2 u z 1 2 Gamma 1 2 h k 1 -1 p 2 c b m -1 2 u z 1 2 2 4 c b m -1 2 u -1 2 -1 p 2 c b m -1 2 u z 1 2 2 c b m -1 2 u -1 1 2 c b m -1 2 u Gamma 1 2 h k 2 -1 p 2 c b m -1 2 u z 1 2 2 4 c b m -1 2 u -1 p 2 -4 c 4 b m -1 8 b u -1 c -1 b m -1 2 u -2 n 1 h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p 2 c -1 b m -1 2 u z 1 2 h k p 2 c -1 b m -1 2 u z 1 2 2 b m -1 2 u -1 c -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p 2 c -1 b m -1 2 u z 1 2 Gamma 1 2 h k 1 p 2 c -1 b m -1 2 u z 1 2 2 4 b m -1 2 u -1 4 c -1 2 p 2 c -1 b m -1 2 u z 1 2 2 b m -1 2 u -1 c -1 1 2 c -1 b m -1 2 u Gamma 1 2 h k 2 p 2 c -1 b m -1 2 u z 1 2 2 4 b m -1 2 u -1 4 c -1 -1 p 2 2 c -1 Binomial m m 2 -1 \$CellContext`m 2 -1 h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n p -1 2 c z 1 2 h k p -1 2 c z 1 2 2 c -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p p -1 2 c z 1 2 Gamma 1 2 h k 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 h k 2 p -1 2 c z 1 2 2 4 c -1 -1 Binomial m m 2 -1 \$CellContext`m 2 -1 h 0 k k 0 n -1 k -1 h 4 k p -1 h -1 k 2 n 2 z 1 2 c p h k -1 2 z 1 2 c p 2 c -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k p 2 z 1 2 c p Gamma 1 2 h k 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 h k 2 -1 2 z 1 2 c p 2 4 c -1 m SuperPlus n [/itex]

 Rule Form

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

 2002-12-18