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 Cosh

 http://functions.wolfram.com/01.20.21.3849.01

 Input Form

 Integrate[E^(p z) Cos[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) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[Binomial[m, s] ((4 E^(p z) Cos[b (m - 2 s) Sqrt[z]])/p + (I b Sqrt[Pi] (m - 2 s) Erfi[((-I) b (m - 2 s) + 2 p Sqrt[z])/ (2 Sqrt[p])])/(E^(((-I) b m + 2 I b s)^2/(4 p)) p^(3/2)) - (I b E^((b^2 (m - 2 s)^2)/(4 p)) Sqrt[Pi] (m - 2 s) Erfi[(I b (m - 2 s) + 2 p Sqrt[z])/(2 Sqrt[p])])/p^(3/2)), {s, 0, Floor[(1/2) (-1 + m)]}] + 2^(-1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, k] ((4 E^(p z) Cosh[c (2 k - v) Sqrt[z]])/ p - (Sqrt[Pi] (-2 c k + c v) Erfi[((-c) (2 k - v) + 2 p Sqrt[z])/ (2 Sqrt[p])])/(E^((-2 c k + c v)^2/(4 p)) p^(3/2)) + (c Sqrt[Pi] (-2 k + v) Erfi[((-c) (-2 k + v) + 2 p Sqrt[z])/ (2 Sqrt[p])])/(E^((c^2 (-2 k + v)^2)/(4 p)) p^(3/2))), {k, 0, Floor[(1/2) (-1 + v)]}] + 2^(-1 - m - v) Sum[Binomial[m, s] Sum[Binomial[v, k] ((4 E^(p z) Cos[(2 I c k + b m - 2 b s - I c v) Sqrt[z]])/p + (4 E^(p z) Cos[(2 I c k - b m + 2 b s - I c v) Sqrt[z]])/p - (1/p^(3/2)) (E^((b (m - 2 s) + I c (-2 k + v))^2/(4 p)) Sqrt[Pi] (b (m - 2 s) + I c (-2 k + v)) Erf[((-b) (m - 2 s) + I c (2 k - v) - 2 I p Sqrt[z])/(2 Sqrt[p])]) + (1/p^(3/2)) (E^((-2 I c k - b m + 2 b s + I c v)^2/(4 p)) Sqrt[Pi] (2 I c k + b m - 2 b s - I c v) Erf[(b (m - 2 s) + I c (2 k - v) - 2 I p Sqrt[z])/(2 Sqrt[p])]) + (1/p^(3/2)) (E^((2 I c k + b m - 2 b s - I c v)^2/(4 p)) Sqrt[Pi] (2 I c k + b m - 2 b s - I c v) Erf[(b (m - 2 s) + I c (2 k - v) + 2 I p Sqrt[z])/(2 Sqrt[p])]) - (1/p^(3/2)) (I E^((b (m - 2 s) + I c (-2 k + v))^2/(4 p)) Sqrt[Pi] (b (m - 2 s) + I c (-2 k + v)) Erfi[(I b (m - 2 s) - c (-2 k + v) + 2 p Sqrt[z])/(2 Sqrt[p])])), {k, 0, Floor[(1/2) (-1 + v)]}], {s, 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 cos m ( b z ) cosh v ( c z ) z 2 - m - v - 1 ( 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]]]]] ( 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]]]]] ( 4 p z cos ( b ( m - 2 s ) z ) p + b - ( 2 b s - b m ) 2 4 p π ( m - 2 s ) erfi ( 2 p z - b ( m - 2 s ) 2 p ) p 3 / 2 - b b 2 ( m - 2 s ) 2 4 p π ( m - 2 s ) erfi ( 2 z p + b ( m - 2 s ) 2 p ) p 3 / 2 ) ) ( 1 - v mod 2 \$CellContext`v 2 ) + 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 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 ) k = 0 v - 1 2 ( v k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 4 p z cosh ( c ( 2 k - v ) z ) p + c - c 2 ( v - 2 k ) 2 4 p π ( v - 2 k ) erfi ( 2 p z - c ( v - 2 k ) 2 p ) p 3 / 2 - - ( c v - 2 c k ) 2 4 p π ( c v - 2 c k ) erfi ( 2 p z - c ( 2 k - v ) 2 p ) p 3 / 2 ) + 2 - m - v - 1 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]]]]] k = 0 v - 1 2 ( v k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 4 p z cos ( ( 2 c k - b m + 2 b s - c v ) z ) p + 4 p z cos ( ( 2 c k + b m - 2 b s - c v ) z ) p + 1 p 3 / 2 ( ( 2 c k + b m - 2 b s - c v ) 2 4 p π ( 2 c k + b m - 2 b s - c v ) erf ( 2 z p + b ( m - 2 s ) + c ( 2 k - v ) 2 p ) ) - 1 p 3 / 2 ( ( b ( m - 2 s ) + c ( v - 2 k ) ) 2 4 p π ( b ( m - 2 s ) + c ( v - 2 k ) ) erf ( - 2 z p - b ( m - 2 s ) + c ( 2 k - v ) 2 p ) ) + 1 p 3 / 2 ( ( - 2 c k - b m + 2 b s + c v ) 2 4 p π ( 2 c k + b m - 2 b s - c v ) erf ( - 2 z p + b ( m - 2 s ) + c ( 2 k - v ) 2 p ) ) - 1 p 3 / 2 ( ( b ( m - 2 s ) + c ( v - 2 k ) ) 2 4 p π ( b ( m - 2 s ) + c ( v - 2 k ) ) erfi ( 2 z p + b ( m - 2 s ) - c ( v - 2 k ) 2 p ) ) ) /; m + v + Condition z p z b z 1 2 m c z 1 2 v 2 -1 m -1 v -1 Binomial v v 2 -1 s 0 m -1 2 -1 Binomial m s 4 p z b m -1 2 s z 1 2 p -1 b -1 2 b s -1 b m 2 4 p -1 1 2 m -1 2 s Erfi 2 p z 1 2 -1 b m -1 2 s 2 p 1 2 -1 p 3 2 -1 -1 b b 2 m -1 2 s 2 4 p -1 1 2 m -1 2 s Erfi 2 z 1 2 p b m -1 2 s 2 p 1 2 -1 p 3 2 -1 1 -1 \$CellContext`v 2 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 Binomial m m 2 -1 1 -1 \$CellContext`m 2 k 0 v -1 2 -1 Binomial v k 4 p z c 2 k -1 v z 1 2 p -1 c -1 c 2 v -1 2 k 2 4 p -1 1 2 v -1 2 k Erfi 2 p z 1 2 -1 c v -1 2 k 2 p 1 2 -1 p 3 2 -1 -1 -1 c v -1 2 c k 2 4 p -1 1 2 c v -1 2 c k Erfi 2 p z 1 2 -1 c 2 k -1 v 2 p 1 2 -1 p 3 2 -1 2 -1 m -1 v -1 s 0 m -1 2 -1 Binomial m s k 0 v -1 2 -1 Binomial v k 4 p z 2 c k -1 b m 2 b s -1 c v z 1 2 p -1 4 p z 2 c k b m -1 2 b s -1 c v z 1 2 p -1 1 p 3 2 -1 2 c k b m -1 2 b s -1 c v 2 4 p -1 1 2 2 c k b m -1 2 b s -1 c v Erf 2 z 1 2 p b m -1 2 s c 2 k -1 v 2 p 1 2 -1 -1 1 p 3 2 -1 b m -1 2 s c v -1 2 k 2 4 p -1 1 2 b m -1 2 s c v -1 2 k Erf -2 z 1 2 p -1 b m -1 2 s c 2 k -1 v 2 p 1 2 -1 1 p 3 2 -1 -2 c k -1 b m 2 b s c v 2 4 p -1 1 2 2 c k b m -1 2 b s -1 c v Erf -2 z 1 2 p b m -1 2 s c 2 k -1 v 2 p 1 2 -1 -1 1 p 3 2 -1 b m -1 2 s c v -1 2 k 2 4 p -1 1 2 b m -1 2 s c v -1 2 k Erfi 2 z 1 2 p b m -1 2 s -1 c v -1 2 k 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