html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Cos

 http://functions.wolfram.com/01.07.21.2635.01

 Input Form

 Integrate[E^(b z^2) Sin[b z^2]^m Cos[c z^2]^v, z] == (1/Sqrt[b]) (2^(-1 - m - v) Sqrt[Pi] Binomial[m, m/2] Binomial[v, v/2] Erfi[Sqrt[b] 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] (Erfi[Sqrt[b - I b (2 k - m)] z]/ Sqrt[b - I b (2 k - m)] + (E^(I m Pi) Erfi[Sqrt[b + I b (2 k - m)] z])/Sqrt[b + I b (2 k - m)]))/E^((1/2) I m Pi), {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[b + I c (2 s - v)] z]/ Sqrt[b + I c (2 s - v)] + Erfi[Sqrt[b - 2 I c s + I c v] z]/ Sqrt[b - 2 I c s + I c v]), {s, 0, Floor[(1/2) (-1 + v)]}] + 2^(-1 - m - v) Sqrt[Pi] Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] ((E^((I m Pi)/2) Erfi[Sqrt[b - I b (-2 k + m) - I c (-2 s + v)] z])/ Sqrt[b - I b (-2 k + m) - I c (-2 s + v)] + Erfi[Sqrt[b + I b (-2 k + m) - I c (-2 s + v)] z]/E^((1/2) I m Pi)/ Sqrt[b + I b (-2 k + m) - I c (-2 s + v)] + (E^((I m Pi)/2) Erfi[Sqrt[b - I b (-2 k + m) + I c (-2 s + v)] z])/ Sqrt[b - I b (-2 k + m) + I c (-2 s + v)] + Erfi[Sqrt[b + I b (-2 k + m) + I c (-2 s + v)] z]/E^((1/2) I m Pi)/ Sqrt[b + I b (-2 k + m) + 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

 b z 2 sin m ( b z 2 ) cos v ( c z 2 ) z 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 ( b z ) ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) 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 ) k = 0 m - 1 2 ( - 1 ) k - 1 2 m π ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( m π erfi ( ( 2 k - m ) b + b z ) ( 2 k - m ) b + b + erfi ( b - b ( 2 k - m ) z ) b - b ( 2 k - m ) ) + 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 ( b + c ( 2 s - v ) z ) b + c ( 2 s - v ) + erfi ( b - 2 c s + c v z ) b - 2 c s + c v ) + 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]]]]] ( - 1 2 m π erfi ( ( m - 2 k ) b + b + c ( v - 2 s ) z ) ( m - 2 k ) b + b + c ( v - 2 s ) + m π 2 erfi ( - ( m - 2 k ) b + b + c ( v - 2 s ) z ) - ( m - 2 k ) b + b + c ( v - 2 s ) + - 1 2 m π erfi ( ( m - 2 k ) b + b - c ( v - 2 s ) z ) ( m - 2 k ) b + b - c ( v - 2 s ) + m π 2 erfi ( - ( m - 2 k ) b + b - c ( v - 2 s ) z ) - ( m - 2 k ) b + b - c ( v - 2 s ) ) /; m + v + Condition z b z 2 b z 2 m c z 2 v 2 -1 m -1 v -1 1 2 Binomial m m 2 -1 Binomial v v 2 -1 Erfi b 1 2 z 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 b 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 -1 1 2 m Binomial m k m Erfi 2 k -1 m b b 1 2 z 2 k -1 m b b 1 2 -1 Erfi b -1 b 2 k -1 m 1 2 z b -1 b 2 k -1 m 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 b c 2 s -1 v 1 2 z b c 2 s -1 v 1 2 -1 Erfi b -1 2 c s c v 1 2 z b -1 2 c s c v 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 2 m Erfi m -1 2 k b b c v -1 2 s 1 2 z m -1 2 k b b c v -1 2 s 1 2 -1 m 2 -1 Erfi -1 m -1 2 k b b c v -1 2 s 1 2 z -1 m -1 2 k b b c v -1 2 s 1 2 -1 -1 1 2 m Erfi m -1 2 k b b -1 c v -1 2 s 1 2 z m -1 2 k b b -1 c v -1 2 s 1 2 -1 m 2 -1 Erfi -1 m -1 2 k b b -1 c v -1 2 s 1 2 z -1 m -1 2 k b b -1 c v -1 2 s 1 2 -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18