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 Cosh

 http://functions.wolfram.com/01.20.21.3839.01

 Input Form

 Integrate[E^(p Sqrt[z]) Cos[b Sqrt[z]]^m Cosh[c z]^v, z] == (1/p^2) (2^(1 - m - v) E^(p Sqrt[z]) (-1 + p Sqrt[z]) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2])) + 2^(-1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, s] (-((2 E^(p Sqrt[z] - c (-2 s + v) z))/ (c (-2 s + v))) + (2 E^(p Sqrt[z] + c (-2 s + v) z))/ (c (-2 s + v)) - (E^(p^2/(4 c (-2 s + v))) p Sqrt[Pi] Erfi[(p - 2 c (-2 s + v) Sqrt[z])/(2 Sqrt[(-c) (-2 s + v)])])/ ((-c) (-2 s + v))^(3/2) - (p Sqrt[Pi] Erfi[(p + 2 c (-2 s + v) Sqrt[z])/(2 Sqrt[c (-2 s + v)])])/ (E^(p^2/(4 c (-2 s + v))) (c (-2 s + v))^(3/2))), {s, 0, Floor[(1/2) (-1 + v)]}] + 2^(2 - m - v) E^(p Sqrt[z]) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(Binomial[m, s] ((-p^2 + b^2 (m - 2 s)^2 (1 + p Sqrt[z]) + p^3 Sqrt[z]) Cos[b (m - 2 s) Sqrt[z]] - b (m - 2 s) (2 p - p^2 Sqrt[z] - b^2 (m - 2 s)^2 Sqrt[z]) Sin[b (m - 2 s) Sqrt[z]]))/((-p + I b (m - 2 s))^2 (p + I b (m - 2 s))^2), {s, 0, Floor[(1/2) (-1 + m)]}] + 2^(-1 - m - v) Sum[Binomial[m, s] Sum[Binomial[v, k] ((2 (E^((p - I b (m - 2 s)) Sqrt[z]) + E^((p + I b (m - 2 s)) Sqrt[z])) (E^(4 c k z) - E^(2 c v z)))/(E^(c (2 k + v) z) (c (2 k - v))) + (E^(((-I) b m + p + 2 I b s)^2/(c (8 k - 4 v))) Sqrt[Pi] (-p + I b (m - 2 s)) Erfi[(-p + I b (m - 2 s) + 2 c (2 k - v) Sqrt[z])/(2 Sqrt[c (-2 k + v)])])/ (c (2 k - v) Sqrt[c (-2 k + v)]) - (Sqrt[Pi] (p + I b (m - 2 s)) Erfi[(p + I b (m - 2 s) + 2 c (2 k - v) Sqrt[z])/(2 Sqrt[c (2 k - v)])])/ E^((p + I b (m - 2 s))^2/(c (8 k - 4 v)))/(c (2 k - v))^(3/2) - (Sqrt[Pi] ((-I) b m + p + 2 I b s) Erfi[((-I) b m + p + 2 I b s + 2 c (2 k - v) Sqrt[z])/(2 Sqrt[c (2 k - v)])])/ E^(((-I) b m + p + 2 I b s)^2/(c (8 k - 4 v)))/ (c (2 k - v))^(3/2) + (E^((p + I b (m - 2 s))^2/(c (8 k - 4 v))) Sqrt[Pi] (p + I b (m - 2 s)) Erfi[(p + I b (m - 2 s) + 2 c (-2 k + v) Sqrt[z])/(2 Sqrt[c (-2 k + v)])])/ (c (2 k - v) Sqrt[c (-2 k + v)])), {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 + 2 p z ( 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]]]]] ( ( z p 3 - p 2 + b 2 ( m - 2 s ) 2 ( z p + 1 ) ) cos ( b ( m - 2 s ) z ) - b ( m - 2 s ) ( - z p 2 + 2 p - b 2 ( m - 2 s ) 2 z ) sin ( b ( m - 2 s ) z ) ) ) / ( ( b ( m - 2 s ) - p ) 2 ( p + b ( m - 2 s ) ) 2 ) ) ( 1 - v mod 2 \$CellContext`v 2 ) + 2 - m - v + 1 p z ( p z - 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]]]]] ( 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 + 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]]]]] ( - p 2 4 c ( v - 2 s ) p π erfi ( p - 2 c ( v - 2 s ) z 2 - c ( v - 2 s ) ) ( - c ( v - 2 s ) ) 3 / 2 + 2 z p + c ( v - 2 s ) z c ( v - 2 s ) - 2 p z - c ( v - 2 s ) z c ( v - 2 s ) - - p 2 4 c ( v - 2 s ) p π erfi ( p + 2 c ( v - 2 s ) z 2 c ( v - 2 s ) ) ( c ( v - 2 s ) ) 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]]]]] ( 2 - c ( 2 k + v ) z ( ( p - b ( m - 2 s ) ) z + ( p + b ( m - 2 s ) ) z ) ( 4 c k z - 2 c v z ) c ( 2 k - v ) + ( - b m + p + 2 b s ) 2 c ( 8 k - 4 v ) π ( b ( m - 2 s ) - p ) erfi ( - p + b ( m - 2 s ) + 2 c ( 2 k - v ) z 2 c ( v - 2 k ) ) c ( 2 k - v ) c ( v - 2 k ) + ( p + b ( m - 2 s ) ) 2 c ( 8 k - 4 v ) π ( p + b ( m - 2 s ) ) erfi ( p + b ( m - 2 s ) + 2 c ( v - 2 k ) z 2 c ( v - 2 k ) ) c ( 2 k - v ) c ( v - 2 k ) - 1 ( c ( 2 k - v ) ) 3 / 2 ( - ( - b m + p + 2 b s ) 2 c ( 8 k - 4 v ) π ( - b m + p + 2 b s ) erfi ( - b m + p + 2 b s + 2 c ( 2 k - v ) z 2 c ( 2 k - v ) ) ) - - ( p + b ( m - 2 s ) ) 2 c ( 8 k - 4 v ) π ( p + b ( m - 2 s ) ) erfi ( p + b ( m - 2 s ) + 2 c ( 2 k - v ) z 2 c ( 2 k - v ) ) ( c ( 2 k - v ) ) 3 / 2 ) /; m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m > 0 v TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] v > 0 Condition z p z 1 2 b z 1 2 m c z v 2 -1 m -1 v 2 p z 1 2 Binomial v v 2 -1 s 0 m -1 2 -1 Binomial m s z 1 2 p 3 -1 p 2 b 2 m -1 2 s 2 z 1 2 p 1 b m -1 2 s z 1 2 -1 b m -1 2 s -1 z 1 2 p 2 2 p -1 b 2 m -1 2 s 2 z 1 2 b m -1 2 s z 1 2 b m -1 2 s -1 p 2 p b m -1 2 s 2 -1 1 -1 \$CellContext`v 2 2 -1 m -1 v 1 p z 1 2 p z 1 2 -1 Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 p 2 -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 p 2 4 c v -1 2 s -1 p 1 2 Erfi p -1 2 c v -1 2 s z 1 2 2 -1 c v -1 2 s 1 2 -1 -1 c v -1 2 s 3 2 -1 2 z 1 2 p c v -1 2 s z c v -1 2 s -1 -1 2 p z 1 2 -1 c v -1 2 s z c v -1 2 s -1 -1 -1 p 2 4 c v -1 2 s -1 p 1 2 Erfi p 2 c v -1 2 s z 1 2 2 c v -1 2 s 1 2 -1 c v -1 2 s 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 2 -1 c 2 k v z p -1 b m -1 2 s z 1 2 p b m -1 2 s z 1 2 4 c k z -1 2 c v z c 2 k -1 v -1 -1 b m p 2 b s 2 c 8 k -1 4 v -1 1 2 b m -1 2 s -1 p Erfi -1 p b m -1 2 s 2 c 2 k -1 v z 1 2 2 c v -1 2 k 1 2 -1 c 2 k -1 v c v -1 2 k 1 2 -1 p b m -1 2 s 2 c 8 k -1 4 v -1 1 2 p b m -1 2 s Erfi p b m -1 2 s 2 c v -1 2 k z 1 2 2 c v -1 2 k 1 2 -1 c 2 k -1 v c v -1 2 k 1 2 -1 -1 1 c 2 k -1 v 3 2 -1 -1 -1 b m p 2 b s 2 c 8 k -1 4 v -1 1 2 -1 b m p 2 b s Erfi -1 b m p 2 b s 2 c 2 k -1 v z 1 2 2 c 2 k -1 v 1 2 -1 -1 -1 p b m -1 2 s 2 c 8 k -1 4 v -1 1 2 p b m -1 2 s Erfi p b m -1 2 s 2 c 2 k -1 v z 1 2 2 c 2 k -1 v 1 2 -1 c 2 k -1 v 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