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 Cosh

 http://functions.wolfram.com/01.20.21.3843.01

 Input Form

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

 Rule Form

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

 2002-12-18