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 Cosh

 http://functions.wolfram.com/01.20.21.3783.01

 Input Form

 Integrate[E^(p z^2) Sin[b z^2]^m Cosh[c z^2]^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])) + 2^(-1 - m - v) Sqrt[Pi] Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k Binomial[m, k] ((E^((I m Pi)/2) Erfi[Sqrt[(-I) b (-2 k + m) + p] z])/ Sqrt[(-I) b (-2 k + m) + p] + Erfi[Sqrt[I b (-2 k + m) + p] z]/ (E^((1/2) I m Pi) Sqrt[I b (-2 k + m) + p])), {k, 0, Floor[(1/2) (-1 + m)]}] + 2^(-1 - m - v) Sqrt[Pi] Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, s] (Erfi[Sqrt[p - c (-2 s + v)] z]/ Sqrt[p - c (-2 s + v)] + Erfi[Sqrt[p + c (-2 s + v)] z]/ Sqrt[p + c (-2 s + v)]), {s, 0, Floor[(1/2) (-1 + v)]}] + 2^(-1 - m - v) Sqrt[Pi] Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (Cos[(m Pi)/2] (Erfi[Sqrt[2 I b k - I b m + p + 2 c s - c v] z]/ Sqrt[2 I b k - I b m + p + 2 c s - c v] + Erfi[Sqrt[-2 I b k + I b m + p + 2 c s - c v] z]/ Sqrt[-2 I b k + I b m + p + 2 c s - c v] + Erfi[Sqrt[2 I b k - I b m + p - 2 c s + c v] z]/ Sqrt[2 I b k - I b m + p - 2 c s + c v] + Erfi[Sqrt[-2 I b k + I b m + p - 2 c s + c v] z]/ Sqrt[-2 I b k + I b m + p - 2 c s + c v]) + I (Erfi[Sqrt[2 I b k - I b m + p + 2 c s - c v] z]/ Sqrt[2 I b k - I b m + p + 2 c s - c v] - Erfi[Sqrt[-2 I b k + I b m + p + 2 c s - c v] z]/ Sqrt[-2 I b k + I b m + p + 2 c s - c v] + Erfi[Sqrt[2 I b k - I b m + p - 2 c s + c v] z]/ Sqrt[2 I b k - I b m + p - 2 c s + c v] - Erfi[Sqrt[-2 I b k + I b m + p - 2 c s + c v] z]/ Sqrt[-2 I b k + I b m + p - 2 c s + c v]) Sin[(m Pi)/2]), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}] /; Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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"\[Pi]"]], "2"], "]"]]]]]], ")"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 p z 2 sin m ( b z 2 ) cosh v ( c z 2 ) 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]]]]] ( k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( m π 2 erfi ( p - b ( m - 2 k ) z ) p - b ( m - 2 k ) + - 1 2 m π erfi ( b ( m - 2 k ) + p z ) b ( m - 2 k ) + p ) ) ( 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 + 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]]]]] ( erfi ( p + c ( v - 2 s ) z ) p + c ( v - 2 s ) + erfi ( p - c ( v - 2 s ) z ) p - c ( v - 2 s ) ) + 2 - m - v - 1 π k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[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]]]]] ( ( erfi ( - 2 b k + b m + p - 2 c s + c v z ) - 2 b k + b m + p - 2 c s + c v + erfi ( 2 b k - b m + p - 2 c s + c v z ) 2 b k - b m + p - 2 c s + c v + erfi ( - 2 b k + b m + p + 2 c s - c v z ) - 2 b k + b m + p + 2 c s - c v + erfi ( 2 b k - b m + p + 2 c s - c v z ) 2 b k - b m + p + 2 c s - c v ) cos ( m π 2 ) + ( - erfi ( - 2 b k + b m + p - 2 c s + c v z ) - 2 b k + b m + p - 2 c s + c v + erfi ( 2 b k - b m + p - 2 c s + c v z ) 2 b k - b m + p - 2 c s + c v + erfi ( 2 b k - b m + p + 2 c s - c v z ) 2 b k - b m + p + 2 c s - c v - erfi ( - 2 b k + b m + p + 2 c s - c v z ) - 2 b k + b m + p + 2 c s - c v ) sin ( m π 2 ) ) /; 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 2 v 2 -1 m -1 v -1 1 2 Binomial v v 2 -1 k 0 m -1 2 -1 -1 k Binomial m k m 2 -1 Erfi p -1 b m -1 2 k 1 2 z p -1 b m -1 2 k 1 2 -1 -1 1 2 m Erfi b m -1 2 k p 1 2 z b m -1 2 k p 1 2 -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 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 Erfi p c v -1 2 s 1 2 z p c v -1 2 s 1 2 -1 Erfi p -1 c v -1 2 s 1 2 z p -1 c v -1 2 s 1 2 -1 2 -1 m -1 v -1 1 2 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s Erfi -2 b k b m p -1 2 c s c v 1 2 z -2 b k b m p -1 2 c s c v 1 2 -1 Erfi 2 b k -1 b m p -1 2 c s c v 1 2 z 2 b k -1 b m p -1 2 c s c v 1 2 -1 Erfi -2 b k b m p 2 c s -1 c v 1 2 z -2 b k b m p 2 c s -1 c v 1 2 -1 Erfi 2 b k -1 b m p 2 c s -1 c v 1 2 z 2 b k -1 b m p 2 c s -1 c v 1 2 -1 m 2 -1 -1 Erfi -2 b k b m p -1 2 c s c v 1 2 z -2 b k b m p -1 2 c s c v 1 2 -1 Erfi 2 b k -1 b m p -1 2 c s c v 1 2 z 2 b k -1 b m p -1 2 c s c v 1 2 -1 Erfi 2 b k -1 b m p 2 c s -1 c v 1 2 z 2 b k -1 b m p 2 c s -1 c v 1 2 -1 -1 Erfi -2 b k b m p 2 c s -1 c v 1 2 z -2 b k b m p 2 c s -1 c v 1 2 -1 m 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