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 Cosh

 http://functions.wolfram.com/01.20.21.3848.01

 Input Form

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