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 Cosh

 http://functions.wolfram.com/01.20.21.1393.01

 Input Form

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

 Rule Form

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

 2002-12-18