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 Cos

 http://functions.wolfram.com/01.07.21.2630.01

 Input Form

 Integrate[E^(p z^2) Sin[b z]^m Cos[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])) + (1/Sqrt[p]) (2^(-1 - m - v) Sqrt[Pi] Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k Binomial[m, k] (Erfi[((-I) b (-2 k + m) + 2 p z)/(2 Sqrt[p])]/ E^(((-b^2) (-2 k + m)^2 - 2 I m p Pi)/(4 p)) + Erfi[(I b (-2 k + m) + 2 p z)/(2 Sqrt[p])]/ E^(((-b^2) (-2 k + m)^2 + 2 I m p Pi)/(4 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] ((p - I c (-2 s + v)) Sqrt[p + I c (-2 s + v)] Erfi[Sqrt[p + I c (-2 s + v)] z] + Sqrt[p - I c (-2 s + v)] (p + I c (-2 s + v)) Erfi[(2 p z - 2 I c (-2 s + v) z)/ (2 Sqrt[p - I c (-2 s + v)])]))/((p - I c (-2 s + v)) (p + I 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] (((Sqrt[p - I c (-2 s + v)] (p + I c (-2 s + v)) Erfi[((-I) b (2 k - m) + 2 p z - 2 I c (-2 s + v) z)/(2 Sqrt[p - I c (-2 s + v)])])/E^(((-b^2) (2 k - m)^2 + 2 I m Pi (p - I c (-2 s + v)))/(4 (p - I c (-2 s + v)))) + ((p - I c (-2 s + v)) Sqrt[p + I c (-2 s + v)] Erfi[(I b (2 k - m) + 2 (p + I c (-2 s + v)) z)/(2 Sqrt[p + I c (-2 s + v)])])/E^(((-b^2) (2 k - m)^2 - 2 I m Pi (p + I c (-2 s + v)))/(4 (p + I c (-2 s + v)))))/ ((p - I c (-2 s + v)) (p + I c (-2 s + v))) + ((Sqrt[p - I c (-2 s + v)] (p + I c (-2 s + v)) Erfi[((-I) b (-2 k + m) + 2 p z - 2 I c (-2 s + v) z)/(2 Sqrt[p - I c (-2 s + v)])])/E^(((-b^2) (-2 k + m)^2 - 2 I m Pi (p - I c (-2 s + v)))/(4 (p - I c (-2 s + v)))) + ((p - I c (-2 s + v)) Sqrt[p + I c (-2 s + v)] Erfi[(I b (-2 k + m) + 2 (p + I c (-2 s + v)) z)/(2 Sqrt[p + I c (-2 s + v)])])/E^(((-b^2) (-2 k + m)^2 + 2 I m Pi (p + I c (-2 s + v)))/(4 (p + I c (-2 s + v)))))/ ((p - I c (-2 s + v)) (p + I c (-2 s + v)))), {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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 MathML Form

 p z 2 sin m ( b z ) cos v ( c z 2 ) z 2 - m - v - 1 π erfi ( p z ) ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) p ( 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]]]]] + 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]]]]] ( ( p + c ( v - 2 s ) ) p - c ( v - 2 s ) erfi ( 2 p z - 2 c ( v - 2 s ) z 2 p - c ( v - 2 s ) ) + p + c ( v - 2 s ) ( p - c ( v - 2 s ) ) erfi ( p + c ( v - 2 s ) z ) ) ) / ( ( p - c ( v - 2 s ) ) ( p + c ( v - 2 s ) ) ) + 1 p ( 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 ) 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]]]]] ( - - b 2 ( m - 2 k ) 2 - 2 π m p 4 p erfi ( 2 p z - b ( m - 2 k ) 2 p ) + - 2 π m p - b 2 ( m - 2 k ) 2 4 p erfi ( b ( m - 2 k ) + 2 p z 2 p ) ) ) + 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]]]]] ( ( - - b 2 ( 2 k - m ) 2 - 2 m π ( p + c ( v - 2 s ) ) 4 ( p + c ( v - 2 s ) ) p + c ( v - 2 s ) ( p - c ( v - 2 s ) ) erfi ( b ( 2 k - m ) + 2 ( p + c ( v - 2 s ) ) z 2 p + c ( v - 2 s ) ) + - 2 m π ( p - c ( v - 2 s ) ) - b 2 ( 2 k - m ) 2 4 ( p - c ( v - 2 s ) ) ( p + c ( v - 2 s ) ) p - c ( v - 2 s ) erfi ( - b ( 2 k - m ) + 2 p z - 2 c ( v - 2 s ) z 2 p - c ( v - 2 s ) ) ) / ( ( p - c ( v - 2 s ) ) ( p + c ( v - 2 s ) ) ) + ( - 2 m π ( p + c ( v - 2 s ) ) - b 2 ( m - 2 k ) 2 4 ( p + c ( v - 2 s ) ) p + c ( v - 2 s ) ( p - c ( v - 2 s ) ) erfi ( b ( m - 2 k ) + 2 ( p + c ( v - 2 s ) ) z 2 p + c ( v - 2 s ) ) + - - b 2 ( m - 2 k ) 2 - 2 m π ( p - c ( v - 2 s ) ) 4 ( p - c ( v - 2 s ) ) ( p + c ( v - 2 s ) ) p - c ( v - 2 s ) erfi ( - b ( m - 2 k ) + 2 p z - 2 c ( v - 2 s ) z 2 p - c ( v - 2 s ) ) ) / ( ( p - c ( v - 2 s ) ) ( p + c ( v - 2 s ) ) ) ) /; m + v + Condition z p z 2 b z m c z 2 v 2 -1 m -1 v -1 1 2 Erfi p 1 2 z 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 p 1 2 -1 Binomial m m 2 -1 Binomial v v 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 p c v -1 2 s p -1 c v -1 2 s 1 2 Erfi 2 p z -1 2 c v -1 2 s z 2 p -1 c v -1 2 s 1 2 -1 p c v -1 2 s 1 2 p -1 c v -1 2 s Erfi p c v -1 2 s 1 2 z p -1 c v -1 2 s p c v -1 2 s -1 1 p 1 2 -1 2 -1 m -1 v -1 1 2 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k Binomial m k -1 -1 b 2 m -1 2 k 2 -1 2 m p 4 p -1 Erfi 2 p z -1 b m -1 2 k 2 p 1 2 -1 -1 2 m p -1 b 2 m -1 2 k 2 4 p -1 Erfi b m -1 2 k 2 p z 2 p 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 -1 -1 b 2 2 k -1 m 2 -1 2 m p c v -1 2 s 4 p c v -1 2 s -1 p c v -1 2 s 1 2 p -1 c v -1 2 s Erfi b 2 k -1 m 2 p c v -1 2 s z 2 p c v -1 2 s 1 2 -1 -1 2 m p -1 c v -1 2 s -1 b 2 2 k -1 m 2 4 p -1 c v -1 2 s -1 p c v -1 2 s p -1 c v -1 2 s 1 2 Erfi -1 b 2 k -1 m 2 p z -1 2 c v -1 2 s z 2 p -1 c v -1 2 s 1 2 -1 p -1 c v -1 2 s p c v -1 2 s -1 -1 2 m p c v -1 2 s -1 b 2 m -1 2 k 2 4 p c v -1 2 s -1 p c v -1 2 s 1 2 p -1 c v -1 2 s Erfi b m -1 2 k 2 p c v -1 2 s z 2 p c v -1 2 s 1 2 -1 -1 -1 b 2 m -1 2 k 2 -1 2 m p -1 c v -1 2 s 4 p -1 c v -1 2 s -1 p c v -1 2 s p -1 c v -1 2 s 1 2 Erfi -1 b m -1 2 k 2 p z -1 2 c v -1 2 s z 2 p -1 c v -1 2 s 1 2 -1 p -1 c v -1 2 s p c v -1 2 s -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18