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 Cosh

 http://functions.wolfram.com/01.20.21.3781.01

 Input Form

 Integrate[E^(p z) Sin[b Sqrt[z]]^m Cosh[c Sqrt[z]]^v, z] == (1/p) (2^(-m - v) E^(p z) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2])) + 2^(-1 - m - v) I^m Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k Binomial[m, k] ((4 E^((-(1/2)) I m Pi + p z) Cos[(m Pi)/2 + b (2 k - m) Sqrt[z]])/p + Sqrt[Pi] (-(((2 I b k - I b m) Erfi[(I b (2 k - m) + 2 p Sqrt[z])/ (2 Sqrt[p])])/(E^((2 I b k - I b m)^2/(4 p)) p^(3/2))) - (I b E^((b^2 (-2 k + m)^2)/(4 p) - I m Pi) (-2 k + m) Erfi[(I b (-2 k + m) + 2 p Sqrt[z])/(2 Sqrt[p])])/p^(3/2))), {k, 0, Floor[(1/2) (-1 + m)]}] + 2^(-1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, s] ((4 E^(p z) Cosh[c (-2 s + v) Sqrt[z]])/p + Sqrt[Pi] (-(((-2 c s + c v) Erfi[((-c) (2 s - v) + 2 p Sqrt[z])/ (2 Sqrt[p])])/(E^((-2 c s + c v)^2/(4 p)) p^(3/2))) + (c (-2 s + v) Erfi[((-c) (-2 s + v) + 2 p Sqrt[z])/(2 Sqrt[p])])/ (E^((c^2 (-2 s + v)^2)/(4 p)) p^(3/2)))), {s, 0, Floor[(1/2) (-1 + v)]}] + 2^(-1 - m - v) Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (((4 E^((I m Pi)/2 + p z) Cos[(m Pi)/2 + (2 b k - b m - 2 I c s + I c v) Sqrt[z]])/p - (1/p^(3/2)) ((Sqrt[Pi] (-2 I b k + I b m - 2 c s + c v) Erfi[(I b (-2 k + m) - c (2 s - v) + 2 p Sqrt[z])/(2 Sqrt[p])])/ E^((-2 I b k + I b m - 2 c s + c v)^2/(4 p))) - (1/p^(3/2)) (E^(I m Pi - (2 I b k - I b m + 2 c s - c v)^2/(4 p)) Sqrt[Pi] (2 I b k - I b m + 2 c s - c v) Erfi[(I b (2 k - m) - c (-2 s + v) + 2 p Sqrt[z])/(2 Sqrt[p])]))/ E^((1/2) I m Pi) + E^((I m Pi)/2) ((1/p) (4 E^((-(1/2)) I m Pi + p z) Cos[(m Pi)/2 + (2 b k - b m + 2 I c s - I c v) Sqrt[z]]) - (1/p^(3/2)) ((Sqrt[Pi] (2 I b k - I b m - 2 c s + c v) Erfi[(I b (2 k - m) - c (2 s - v) + 2 p Sqrt[z])/(2 Sqrt[p])])/ E^((2 I b k - I b m - 2 c s + c v)^2/(4 p))) - (1/p^(3/2)) (E^((-I) m Pi - (I b (-2 k + m) - c (-2 s + v))^2/ (4 p)) Sqrt[Pi] (I b (-2 k + m) - c (-2 s + v)) Erfi[(I b (-2 k + m) - c (-2 s + v) + 2 p Sqrt[z])/(2 Sqrt[p])]))), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 p z sin m ( b z ) cosh v ( c z ) z 2 - m - v p z ( 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]]]]] ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) p + 2 - m - v - 1 m ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox[FractionBox["v", "2"], Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 1 - v mod 2 \$CellContext`v 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]]]]] ( π ( - - ( 2 b k - b m ) 2 4 p ( 2 b k - b m ) erfi ( b ( 2 k - m ) + 2 p z 2 p ) p 3 / 2 - b b 2 ( m - 2 k ) 2 4 p - m π ( m - 2 k ) erfi ( b ( m - 2 k ) + 2 p z 2 p ) p 3 / 2 ) + 4 p z - m π 2 cos ( b z ( 2 k - m ) + m π 2 ) p ) + 2 - m - v - 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( π ( c - c 2 ( v - 2 s ) 2 4 p ( v - 2 s ) erfi ( 2 p z - c ( v - 2 s ) 2 p ) p 3 / 2 - - ( c v - 2 c s ) 2 4 p ( c v - 2 c s ) erfi ( 2 p z - c ( 2 s - v ) 2 p ) p 3 / 2 ) + 4 p z cosh ( c ( v - 2 s ) z ) p ) + 2 - m - v - 1 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]]]]] s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - 1 2 m π ( 4 π m 2 + p z cos ( π m 2 + ( 2 b k - b m - 2 c s + c v ) z ) p - 1 p 3 / 2 ( - ( - 2 b k + b m - 2 c s + c v ) 2 4 p π ( - 2 b k + b m - 2 c s + c v ) erfi ( b ( m - 2 k ) - c ( 2 s - v ) + 2 p z 2 p ) ) - 1 p 3 / 2 ( m π - ( 2 b k - b m + 2 c s - c v ) 2 4 p π ( 2 b k - b m + 2 c s - c v ) erfi ( b ( 2 k - m ) - c ( v - 2 s ) + 2 p z 2 p ) ) ) + m π 2 ( 4 p z - m π 2 cos ( π m 2 + ( 2 b k - b m + 2 c s - c v ) z ) p - 1 p 3 / 2 ( - ( 2 b k - b m - 2 c s + c v ) 2 4 p π ( 2 b k - b m - 2 c s + c v ) erfi ( b ( 2 k - m ) - c ( 2 s - v ) + 2 p z 2 p ) ) - 1 p 3 / 2 ( - ( b ( m - 2 k ) - c ( v - 2 s ) ) 2 4 p - m π π ( b ( m - 2 k ) - c ( v - 2 s ) ) erfi ( b ( m - 2 k ) - c ( v - 2 s ) + 2 p z 2 p ) ) ) ) /; m + v + Condition z p z b z 1 2 m c z 1 2 v 2 -1 m -1 v p z Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 p -1 2 -1 m -1 v -1 m Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k Binomial m k 1 2 -1 -1 2 b k -1 b m 2 4 p -1 2 b k -1 b m Erfi b 2 k -1 m 2 p z 1 2 2 p 1 2 -1 p 3 2 -1 -1 b b 2 m -1 2 k 2 4 p -1 -1 m m -1 2 k Erfi b m -1 2 k 2 p z 1 2 2 p 1 2 -1 p 3 2 -1 4 p z -1 m 2 -1 b z 1 2 2 k -1 m m 2 -1 p -1 2 -1 m -1 v -1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 Binomial v s 1 2 c -1 c 2 v -1 2 s 2 4 p -1 v -1 2 s Erfi 2 p z 1 2 -1 c v -1 2 s 2 p 1 2 -1 p 3 2 -1 -1 -1 c v -1 2 c s 2 4 p -1 c v -1 2 c s Erfi 2 p z 1 2 -1 c 2 s -1 v 2 p 1 2 -1 p 3 2 -1 4 p z c v -1 2 s z 1 2 p -1 2 -1 m -1 v -1 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s -1 1 2 m 4 m 2 -1 p z m 2 -1 2 b k -1 b m -1 2 c s c v z 1 2 p -1 -1 1 p 3 2 -1 -1 -2 b k b m -1 2 c s c v 2 4 p -1 1 2 -2 b k b m -1 2 c s c v Erfi b m -1 2 k -1 c 2 s -1 v 2 p z 1 2 2 p 1 2 -1 -1 1 p 3 2 -1 m -1 2 b k -1 b m 2 c s -1 c v 2 4 p -1 1 2 2 b k -1 b m 2 c s -1 c v Erfi b 2 k -1 m -1 c v -1 2 s 2 p z 1 2 2 p 1 2 -1 m 2 -1 4 p z -1 m 2 -1 m 2 -1 2 b k -1 b m 2 c s -1 c v z 1 2 p -1 -1 1 p 3 2 -1 -1 2 b k -1 b m -1 2 c s c v 2 4 p -1 1 2 2 b k -1 b m -1 2 c s c v Erfi b 2 k -1 m -1 c 2 s -1 v 2 p z 1 2 2 p 1 2 -1 -1 1 p 3 2 -1 -1 b m -1 2 k -1 c v -1 2 s 2 4 p -1 -1 m 1 2 b m -1 2 k -1 c v -1 2 s Erfi b m -1 2 k -1 c v -1 2 s 2 p z 1 2 2 p 1 2 -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18