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; }

 Cosh

 http://functions.wolfram.com/01.20.21.3786.01

 Input Form

 Integrate[E^(b z^2 + e) Sin[a z^2 + q]^m Cosh[c z^2 + g]^v, z] == (1/Sqrt[b]) (2^(-1 - m - v) E^e 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 E^(e - (I m Pi)/2 + I (-2 k + m) q) Binomial[m, k] ((E^(2 ((I m Pi)/2 - I (-2 k + m) q)) Erfi[Sqrt[b - I a (-2 k + m)] z])/ Sqrt[b - I a (-2 k + m)] + Erfi[Sqrt[b + I a (-2 k + m)] z]/ Sqrt[b + I a (-2 k + m)]), {k, 0, Floor[(1/2) (-1 + m)]}] + 2^(-1 - m - v) Sqrt[Pi] Binomial[m, m/2] (1 - Mod[m, 2]) Sum[E^(e - g (-2 k + v)) Binomial[v, k] (Erfi[Sqrt[b - c (-2 k + v)] z]/Sqrt[b - c (-2 k + v)] + (E^(2 g (-2 k + v)) Erfi[Sqrt[b + c (-2 k + v)] z])/ Sqrt[b + c (-2 k + v)]), {k, 0, Floor[(1/2) (-1 + v)]}] + 2^(-1 - m - v) Sqrt[Pi] Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (E^(e + I (-2 k + m) q - (1/2) I Pi m - g (-2 s + v)) (Erfi[Sqrt[b + I a (-2 k + m) - c (-2 s + v)] z]/ Sqrt[b + I a (-2 k + m) - c (-2 s + v)] + (E^(2 ((-I) (-2 k + m) q + (1/2) I Pi m + g (-2 s + v))) Erfi[Sqrt[b - I a (-2 k + m) + c (-2 s + v)] z])/ Sqrt[b - I a (-2 k + m) + c (-2 s + v)]) + E^(e - I (-2 k + m) q + (1/2) I Pi m - g (-2 s + v)) (Erfi[Sqrt[b - I a (-2 k + m) - c (-2 s + v)] z]/ Sqrt[b - I a (-2 k + m) - c (-2 s + v)] + (E^(2 (I (-2 k + m) q - (1/2) I Pi m + g (-2 s + v))) Erfi[Sqrt[b + I a (-2 k + m) + c (-2 s + v)] z])/ Sqrt[b + I a (-2 k + m) + 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 + e sin m ( a z 2 + q ) cosh v ( c z 2 + g ) 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 e + ( m - 2 k ) q - m π 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( erfi ( b + a ( m - 2 k ) z ) b + a ( m - 2 k ) + 2 ( m π 2 - ( m - 2 k ) q ) erfi ( b - a ( m - 2 k ) z ) b - a ( m - 2 k ) ) ) ( 1 - v mod 2 \$CellContext`v 2 ) + 2 - m - v - 1 e 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 ( 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 ) k = 0 v - 1 2 e - g ( v - 2 k ) ( v k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 g ( v - 2 k ) erfi ( b + c ( v - 2 k ) z ) b + c ( v - 2 k ) + erfi ( b - c ( v - 2 k ) z ) b - c ( v - 2 k ) ) + 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]]]]] ( e - ( m - 2 k ) q - g ( v - 2 s ) + m π 2 ( 2 ( - 1 2 π m + ( m - 2 k ) q + g ( v - 2 s ) ) erfi ( b + a ( m - 2 k ) + c ( v - 2 s ) z ) b + a ( m - 2 k ) + c ( v - 2 s ) + erfi ( b - a ( m - 2 k ) - c ( v - 2 s ) z ) b - a ( m - 2 k ) - c ( v - 2 s ) ) + e - π m 2 + ( m - 2 k ) q - g ( v - 2 s ) ( 2 ( π m 2 - ( m - 2 k ) q + g ( v - 2 s ) ) erfi ( b - a ( m - 2 k ) + c ( v - 2 s ) z ) b - a ( m - 2 k ) + c ( v - 2 s ) + erfi ( b + a ( m - 2 k ) - c ( v - 2 s ) z ) b + a ( m - 2 k ) - c ( v - 2 s ) ) ) /; m + v + Condition z b z 2 e a z 2 q m c z 2 g v 2 -1 m -1 v -1 1 2 Binomial v v 2 -1 k 0 m -1 2 -1 -1 k e m -1 2 k q -1 m 2 -1 Binomial m k Erfi b a m -1 2 k 1 2 z b a m -1 2 k 1 2 -1 2 m 2 -1 -1 m -1 2 k q Erfi b -1 a m -1 2 k 1 2 z b -1 a m -1 2 k 1 2 -1 1 -1 \$CellContext`v 2 2 -1 m -1 v -1 e 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 m m 2 -1 1 -1 \$CellContext`m 2 k 0 v -1 2 -1 e -1 g v -1 2 k Binomial v k 2 g v -1 2 k Erfi b c v -1 2 k 1 2 z b c v -1 2 k 1 2 -1 Erfi b -1 c v -1 2 k 1 2 z b -1 c v -1 2 k 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 e -1 m -1 2 k q -1 g v -1 2 s m 2 -1 2 -1 1 2 m m -1 2 k q g v -1 2 s Erfi b a m -1 2 k c v -1 2 s 1 2 z b a m -1 2 k c v -1 2 s 1 2 -1 Erfi b -1 a m -1 2 k -1 c v -1 2 s 1 2 z b -1 a m -1 2 k -1 c v -1 2 s 1 2 -1 e -1 m 2 -1 m -1 2 k q -1 g v -1 2 s 2 m 2 -1 -1 m -1 2 k q g v -1 2 s Erfi b -1 a m -1 2 k c v -1 2 s 1 2 z b -1 a m -1 2 k c v -1 2 s 1 2 -1 Erfi b a m -1 2 k -1 c v -1 2 s 1 2 z b a m -1 2 k -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