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/01.20.21.3846.01

 Input Form

 Integrate[E^(p z^2) Cos[b z]^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])) + (1/Sqrt[p]) (2^(-1 - m - v) Sqrt[Pi] Binomial[v, v/2] (1 - Mod[v, 2]) Sum[Binomial[m, s] (E^((b^2 (m - 2 s)^2)/(4 p)) Erfi[((-I) b (m - 2 s) + 2 p z)/(2 Sqrt[p])] + E^((b^2 (m - 2 s)^2)/(4 p)) Erfi[(I b (m - 2 s) + 2 p z)/ (2 Sqrt[p])]), {s, 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 - c (-2 s + v)) Sqrt[p + c (-2 s + v)] Erfi[Sqrt[p + c (-2 s + v)] z] + Sqrt[p - c (-2 s + v)] (p + c (-2 s + v)) Erfi[(2 p z - 2 c (-2 s + v) z)/ (2 Sqrt[p - c (-2 s + v)])]))/((p - c (-2 s + v)) (p + c (-2 s + v))), {s, 0, Floor[(1/2) (-1 + v)]}] + 2^(-1 - m - v) Sqrt[Pi] Sum[Binomial[m, s] Sum[Binomial[v, k] ((((2 c k + p - c v) Sqrt[-2 c k + p + c v] Erfi[((-I) b m + 2 I b s + 2 p z - 2 (2 c k - c v) z)/(2 Sqrt[-2 c k + p + c v])])/E^(((-I) b m + 2 I b s)^2/ (4 (-2 c k + p + c v))) + (Sqrt[2 c k + p - c v] (-2 c k + p + c v) Erfi[(I b m - 2 I b s + 2 (2 c k + p - c v) z)/(2 Sqrt[2 c k + p - c v])])/E^((I b m - 2 I b s)^2/ (4 (2 c k + p - c v))))/((2 c k + p - c v) (-2 c k + p + c v)) + (((2 c k + p - c v) Sqrt[-2 c k + p + c v] Erfi[(I b m - 2 I b s + 2 p z - 2 (2 c k - c v) z)/(2 Sqrt[-2 c k + p + c v])])/E^((I b m - 2 I b s)^2/ (4 (-2 c k + p + c v))) + (Sqrt[2 c k + p - c v] (-2 c k + p + c v) Erfi[((-I) b m + 2 I b s + 2 (2 c k + p - c v) z)/(2 Sqrt[2 c k + p - c v])])/ E^(((-I) b m + 2 I b s)^2/(4 (2 c k + p - c v))))/ ((2 c k + p - c v) (-2 c k + p + c v))), {k, 0, Floor[(1/2) (-1 + v)]}], {s, 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 cos m ( b z ) cosh 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 ( 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]]]]] ( ( 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 ) s = 0 m - 1 2 ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b 2 ( m - 2 s ) 2 4 p erfi ( 2 p z - b ( m - 2 s ) 2 p ) + b 2 ( m - 2 s ) 2 4 p erfi ( b ( m - 2 s ) + 2 p z 2 p ) ) ) + 2 - m - v - 1 π s = 0 m - 1 2 ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] k = 0 v - 1 2 ( v k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( ( - ( b m - 2 b s ) 2 4 ( - 2 c k + p + c v ) - 2 c k + p + c v ( 2 c k + p - c v ) erfi ( b m - 2 b s + 2 p z - 2 ( 2 c k - c v ) z 2 - 2 c k + p + c v ) + - ( 2 b s - b m ) 2 4 ( 2 c k + p - c v ) ( - 2 c k + p + c v ) 2 c k + p - c v erfi ( - b m + 2 b s + 2 ( 2 c k + p - c v ) z 2 2 c k + p - c v ) ) / ( ( 2 c k + p - c v ) ( - 2 c k + p + c v ) ) + ( - ( 2 b s - b m ) 2 4 ( - 2 c k + p + c v ) - 2 c k + p + c v ( 2 c k + p - c v ) erfi ( - b m + 2 b s + 2 p z - 2 ( 2 c k - c v ) z 2 - 2 c k + p + c v ) + - ( b m - 2 b s ) 2 4 ( 2 c k + p - c v ) ( - 2 c k + p + c v ) 2 c k + p - c v erfi ( b m - 2 b s + 2 ( 2 c k + p - c v ) z 2 2 c k + p - c v ) ) / ( ( 2 c k + p - c v ) ( - 2 c k + p + c v ) ) ) /; m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m > 0 v TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] v > 0 Condition z p z 2 b z m c z 2 v 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 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 s 0 m -1 2 -1 Binomial m s b 2 m -1 2 s 2 4 p -1 Erfi 2 p z -1 b m -1 2 s 2 p 1 2 -1 b 2 m -1 2 s 2 4 p -1 Erfi b m -1 2 s 2 p z 2 p 1 2 -1 2 -1 m -1 v -1 1 2 s 0 m -1 2 -1 Binomial m s k 0 v -1 2 -1 Binomial v k -1 b m -1 2 b s 2 4 -2 c k p c v -1 -2 c k p c v 1 2 2 c k p -1 c v Erfi b m -1 2 b s 2 p z -1 2 2 c k -1 c v z 2 -2 c k p c v 1 2 -1 -1 2 b s -1 b m 2 4 2 c k p -1 c v -1 -2 c k p c v 2 c k p -1 c v 1 2 Erfi -1 b m 2 b s 2 2 c k p -1 c v z 2 2 c k p -1 c v 1 2 -1 2 c k p -1 c v -2 c k p c v -1 -1 2 b s -1 b m 2 4 -2 c k p c v -1 -2 c k p c v 1 2 2 c k p -1 c v Erfi -1 b m 2 b s 2 p z -1 2 2 c k -1 c v z 2 -2 c k p c v 1 2 -1 -1 b m -1 2 b s 2 4 2 c k p -1 c v -1 -2 c k p c v 2 c k p -1 c v 1 2 Erfi b m -1 2 b s 2 2 c k p -1 c v z 2 2 c k p -1 c v 1 2 -1 2 c k p -1 c v -2 c k p c v -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