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/01.20.21.3844.01

 Input Form

 Integrate[E^(p z) Cos[b z]^m Cosh[c z^2]^v, z] == (1/p) (2^(-m - v) E^(p z) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2])) + 2^(-m - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(E^((p - I b (m - 2 s)) z)/(p - I b (m - 2 s)) + E^((p + I b (m - 2 s)) z)/(p + I b (m - 2 s))) Binomial[m, s], {s, 0, Floor[(1/2) (-1 + m)]}] - (1/c) (2^(-1 - m - v) Sqrt[Pi] Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(1/(-2 s + v)) Binomial[v, s] (E^(p^2/(4 c (-2 s + v))) Sqrt[(-c) (-2 s + v)] Erfi[(p - 2 c (-2 s + v) z)/ (2 Sqrt[(-c) (-2 s + v)])] - (Sqrt[c (-2 s + v)] Erfi[(p + 2 c (-2 s + v) z)/(2 Sqrt[c (-2 s + v)])])/ E^(p^2/(4 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 - c v) Sqrt[-2 c k + c v] Erfi[((-I) b m + p + 2 I b s - 2 (2 c k - c v) z)/(2 Sqrt[-2 c k + c v])])/E^(((-I) b m + p + 2 I b s)^2/ (4 (-2 c k + c v))) + (Sqrt[2 c k - c v] (-2 c k + c v) Erfi[(I b m + p - 2 I b s + 2 (2 c k - c v) z)/(2 Sqrt[2 c k - c v])])/E^((I b m + p - 2 I b s)^2/ (4 (2 c k - c v))))/((2 c k - c v) (-2 c k + c v)) + (((2 c k - c v) Sqrt[-2 c k + c v] Erfi[(I b m + p - 2 I b s - 2 (2 c k - c v) z)/(2 Sqrt[-2 c k + c v])])/ E^((I b m + p - 2 I b s)^2/(4 (-2 c k + c v))) + (Sqrt[2 c k - c v] (-2 c k + c v) Erfi[((-I) b m + p + 2 I b s + 2 (2 c k - c v) z)/(2 Sqrt[2 c k - c v])])/ E^(((-I) b m + p + 2 I b s)^2/(4 (2 c k - c v))))/ ((2 c k - c v) (-2 c k + 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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"\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 p z cos m ( b z ) cosh v ( c z 2 ) z 2 - m - v ( 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]]]]] ( s = 0 m - 1 2 ( ( p - b ( m - 2 s ) ) z p - b ( m - 2 s ) + ( p + b ( m - 2 s ) ) z p + b ( m - 2 s ) ) ( m s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ) ( 1 - v mod 2 \$CellContext`v 2 ) + 2 - m - v p z ( 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]]]]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) p - 1 c ( 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 1 v - 2 s ( ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity]], List[TagBox["s", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( p 2 4 c ( v - 2 s ) - c ( v - 2 s ) erfi ( p - 2 c ( v - 2 s ) z 2 - c ( v - 2 s ) ) - - p 2 4 c ( v - 2 s ) c ( v - 2 s ) erfi ( p + 2 c ( v - 2 s ) z 2 c ( v - 2 s ) ) ) ) ) + 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 + p - 2 b s ) 2 4 ( c v - 2 c k ) c v - 2 c k ( 2 c k - c v ) erfi ( b m + p - 2 b s - 2 ( 2 c k - c v ) z 2 c v - 2 c k ) + - ( - b m + p + 2 b s ) 2 4 ( 2 c k - c v ) ( c v - 2 c k ) 2 c k - c v erfi ( - b m + p + 2 b s + 2 ( 2 c k - c v ) z 2 2 c k - c v ) ) / ( ( 2 c k - c v ) ( c v - 2 c k ) ) + ( - ( - b m + p + 2 b s ) 2 4 ( c v - 2 c k ) c v - 2 c k ( 2 c k - c v ) erfi ( - b m + p + 2 b s - 2 ( 2 c k - c v ) z 2 c v - 2 c k ) + - ( b m + p - 2 b s ) 2 4 ( 2 c k - c v ) ( c v - 2 c k ) 2 c k - c v erfi ( b m + p - 2 b s + 2 ( 2 c k - c v ) z 2 2 c k - c v ) ) / ( ( 2 c k - c v ) ( c v - 2 c k ) ) ) /; m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m > 0 v TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] v > 0 Condition z p z b z m c z 2 v 2 -1 m -1 v Binomial v v 2 -1 s 0 m -1 2 -1 p -1 b m -1 2 s z p -1 b m -1 2 s -1 p b m -1 2 s z p b m -1 2 s -1 Binomial m s 1 -1 \$CellContext`v 2 2 -1 m -1 v p z Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 p -1 -1 1 c -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 1 v -1 2 s -1 Binomial v s p 2 4 c v -1 2 s -1 -1 c v -1 2 s 1 2 Erfi p -1 2 c v -1 2 s z 2 -1 c v -1 2 s 1 2 -1 -1 -1 p 2 4 c v -1 2 s -1 c v -1 2 s 1 2 Erfi p 2 c v -1 2 s z 2 c v -1 2 s 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 p -1 2 b s 2 4 c v -1 2 c k -1 c v -1 2 c k 1 2 2 c k -1 c v Erfi b m p -1 2 b s -1 2 2 c k -1 c v z 2 c v -1 2 c k 1 2 -1 -1 -1 b m p 2 b s 2 4 2 c k -1 c v -1 c v -1 2 c k 2 c k -1 c v 1 2 Erfi -1 b m p 2 b s 2 2 c k -1 c v z 2 2 c k -1 c v 1 2 -1 2 c k -1 c v c v -1 2 c k -1 -1 -1 b m p 2 b s 2 4 c v -1 2 c k -1 c v -1 2 c k 1 2 2 c k -1 c v Erfi -1 b m p 2 b s -1 2 2 c k -1 c v z 2 c v -1 2 c k 1 2 -1 -1 b m p -1 2 b s 2 4 2 c k -1 c v -1 c v -1 2 c k 2 c k -1 c v 1 2 Erfi b m p -1 2 b s 2 2 c k -1 c v z 2 2 c k -1 c v 1 2 -1 2 c k -1 c v c v -1 2 c k -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