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/01.20.21.4665.01

 Input Form

 Integrate[E^(b z^2 + e) Sinh[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]))/I^m) + 2^(-1 - m - v) Sqrt[Pi] Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(-1)^k E^(e + (-2 k - m) q) Binomial[m, k] ((E^(2 m q) Erfi[Sqrt[b - a (2 k - m)] z])/Sqrt[b - a (2 k - m)] + ((-1)^m E^(4 k q) Erfi[Sqrt[b + a (2 k - m)] z])/ Sqrt[b + 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 s + v)) Binomial[v, s] (Erfi[Sqrt[b - c (2 s - v)] z]/ (E^(4 g s) Sqrt[b - c (2 s - v)]) + Erfi[Sqrt[b + 2 c s - c v] z]/ (E^(2 g v) Sqrt[b + 2 c s - c v])), {s, 0, Floor[(1/2) (-1 + v)]}])/ I^m + (2^(-1 - m - v) Sqrt[Pi] Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] ((E^(e - (I m Pi)/2 - (-2 k + m) q - g (-2 s + v)) Erfi[Sqrt[b - a (-2 k + m) - c (-2 s + v)] z])/ Sqrt[b - a (-2 k + m) - c (-2 s + v)] + (E^(e + (I m Pi)/2 + (-2 k + m) q - g (-2 s + v)) Erfi[Sqrt[b + a (-2 k + m) - c (-2 s + v)] z])/ Sqrt[b + a (-2 k + m) - c (-2 s + v)] + (E^(e - (I m Pi)/2 - (-2 k + m) q + g (-2 s + v)) Erfi[Sqrt[b - a (-2 k + m) + c (-2 s + v)] z])/ Sqrt[b - a (-2 k + m) + c (-2 s + v)] + (E^(e + (I m Pi)/2 + (-2 k + m) q + g (-2 s + v)) Erfi[Sqrt[b + a (-2 k + m) + c (-2 s + v)] z])/ Sqrt[b + a (-2 k + m) + c (-2 s + v)]), {s, 0, Floor[(1/2) (-1 + v)]}], {k, 0, Floor[(1/2) (-1 + m)]}])/ I^m /; Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 b z 2 + e sinh m ( a z 2 + q ) cosh v ( c z 2 + g ) z 2 - m - v - 1 - m 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 ( 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 e + ( - 2 k - m ) q ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - 1 ) m 4 k q erfi ( b + a ( 2 k - m ) z ) b + a ( 2 k - m ) + 2 m q erfi ( b - a ( 2 k - m ) z ) b - a ( 2 k - m ) ) + 2 - m - v - 1 - m 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 e + g ( 2 s + v ) ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( - 4 g s erfi ( b - c ( 2 s - v ) z ) b - c ( 2 s - v ) + - 2 g v erfi ( b + 2 c s - c v z ) b + 2 c s - c v ) + 2 - m - v - 1 - m 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 erfi ( b + a ( m - 2 k ) + c ( v - 2 s ) z ) b + a ( m - 2 k ) + c ( v - 2 s ) + e - ( m - 2 k ) q + g ( v - 2 s ) - m π 2 erfi ( b - a ( m - 2 k ) + c ( v - 2 s ) z ) b - a ( m - 2 k ) + c ( v - 2 s ) + e + ( m - 2 k ) q - g ( v - 2 s ) + m π 2 erfi ( b + a ( m - 2 k ) - c ( v - 2 s ) z ) b + a ( m - 2 k ) - c ( v - 2 s ) + e - ( m - 2 k ) q - g ( v - 2 s ) - m π 2 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 m 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 v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 -1 k e -2 k -1 m q Binomial m k -1 m 4 k q Erfi b a 2 k -1 m 1 2 z b a 2 k -1 m 1 2 -1 2 m q Erfi b -1 a 2 k -1 m 1 2 z b -1 a 2 k -1 m 1 2 -1 2 -1 m -1 v -1 -1 m 1 2 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 e g 2 s v Binomial v s -4 g s Erfi b -1 c 2 s -1 v 1 2 z b -1 c 2 s -1 v 1 2 -1 -2 g v Erfi b 2 c s -1 c v 1 2 z b 2 c s -1 c v 1 2 -1 2 -1 m -1 v -1 -1 m 1 2 k 0 m -1 2 -1 -1 k Binomial m k s 0 v -1 2 -1 Binomial v s e m -1 2 k q g v -1 2 s m 2 -1 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 e -1 m -1 2 k q g v -1 2 s -1 m 2 -1 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 e m -1 2 k q -1 g v -1 2 s m 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 e -1 m -1 2 k q -1 g v -1 2 s -1 m 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 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18