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 Cosh

 http://functions.wolfram.com/01.20.21.3838.01

 Input Form

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