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/01.20.21.3842.01

 Input Form

 Integrate[E^(p z) Cos[b z^2]^m Cosh[c z]^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])) + (I/b) (2^(-1 - m - v) Sqrt[Pi] Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(1/(m - 2 s)) Binomial[m, s] ((Sqrt[(-I) b (m - 2 s)] Erfi[(p - 2 I b (m - 2 s) z)/ (2 Sqrt[(-I) b (m - 2 s)])])/E^((I p^2)/(4 b (m - 2 s))) - E^((I p^2)/(4 b (m - 2 s))) Sqrt[I b (m - 2 s)] Erfi[(p + 2 I b (m - 2 s) z)/(2 Sqrt[I b (m - 2 s)])]), {s, 0, Floor[(1/2) (-1 + m)]}]) + 2^(1 - m - v) E^(p z) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(Binomial[v, s] ((-p) Cosh[c (-2 s + v) z] + c (-2 s + v) Sinh[c (-2 s + v) z]))/((-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] ((((I b m - 2 I b s) Sqrt[(-I) b m + 2 I b s] Erfi[(-2 c k + p + c v - 2 (I b m - 2 I b s) z)/(2 Sqrt[(-I) b m + 2 I b s])])/E^((-2 c k + p + c v)^2/ (4 ((-I) b m + 2 I b s))) + (Sqrt[I b m - 2 I b s] ((-I) b m + 2 I b s) Erfi[(2 c k + p - c v + 2 (I b m - 2 I b s) z)/(2 Sqrt[I b m - 2 I b s])])/E^((2 c k + p - c v)^2/ (4 (I b m - 2 I b s))))/(4 (I b m - 2 I b s) ((-I) b m + 2 I b s)) + ((Sqrt[I b m - 2 I b s] ((-I) b m + 2 I b s) Erfi[(-2 c k + p + c v - 2 ((-I) b m + 2 I b s) z)/(2 Sqrt[I b m - 2 I b s])])/E^((-2 c k + p + c v)^2/ (4 (I b m - 2 I b s))) + ((I b m - 2 I b s) Sqrt[(-I) b m + 2 I b s] Erfi[(2 c k + p - c v + 2 ((-I) b m + 2 I b s) z)/(2 Sqrt[(-I) b m + 2 I b s])])/ E^((2 c k + p - c v)^2/(4 ((-I) b m + 2 I b s))))/ (4 (I b m - 2 I b s) ((-I) b m + 2 I b s))), {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 cos m ( b z 2 ) cosh v ( c z ) z 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 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 ) s = 0 m - 1 2 1 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]]]]] ( - p 2 4 b ( m - 2 s ) - b ( m - 2 s ) erfi ( p - 2 b ( m - 2 s ) z 2 - b ( m - 2 s ) ) - p 2 4 b ( m - 2 s ) b ( m - 2 s ) erfi ( p + 2 b ( m - 2 s ) z 2 b ( m - 2 s ) ) ) ) ) ) + 2 - m - v + 1 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]]]]] ( 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]]]]] ( c ( v - 2 s ) sinh ( c ( v - 2 s ) z ) - p cosh ( c ( v - 2 s ) z ) ) ( c ( v - 2 s ) - p ) ( p + 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]]]]] ( ( - ( 2 c k + p - c v ) 2 4 ( 2 b s - b m ) 2 b s - b m ( b m - 2 b s ) erfi ( 2 c k + p - c v + 2 ( 2 b s - b m ) z 2 2 b s - b m ) + - ( - 2 c k + p + c v ) 2 4 ( b m - 2 b s ) ( 2 b s - b m ) b m - 2 b s erfi ( - 2 c k + p + c v - 2 ( 2 b s - b m ) z 2 b m - 2 b s ) ) / ( 4 ( b m - 2 b s ) ( 2 b s - b m ) ) + ( - ( - 2 c k + p + c v ) 2 4 ( 2 b s - b m ) 2 b s - b m ( b m - 2 b s ) erfi ( - 2 c k + p + c v - 2 ( b m - 2 b s ) z 2 2 b s - b m ) + - ( 2 c k + p - c v ) 2 4 ( b m - 2 b s ) ( 2 b s - b m ) b m - 2 b s erfi ( 2 c k + p - c v + 2 ( b m - 2 b s ) z 2 b m - 2 b s ) ) / ( 4 ( b m - 2 b s ) ( 2 b s - b m ) ) ) /; m TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] m > 0 v TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] v > 0 Condition z p z b z 2 m c z v 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 b -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 1 m -1 2 s -1 Binomial m s -1 p 2 4 b m -1 2 s -1 -1 b m -1 2 s 1 2 Erfi p -1 2 b m -1 2 s z 2 -1 b m -1 2 s 1 2 -1 -1 p 2 4 b m -1 2 s -1 b m -1 2 s 1 2 Erfi p 2 b m -1 2 s z 2 b m -1 2 s 1 2 -1 2 -1 m -1 v 1 p z Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 Binomial v s c v -1 2 s c v -1 2 s z -1 p c v -1 2 s z c v -1 2 s -1 p p c v -1 2 s -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 2 c k p -1 c v 2 4 2 b s -1 b m -1 2 b s -1 b m 1 2 b m -1 2 b s Erfi 2 c k p -1 c v 2 2 b s -1 b m z 2 2 b s -1 b m 1 2 -1 -1 -2 c k p c v 2 4 b m -1 2 b s -1 2 b s -1 b m b m -1 2 b s 1 2 Erfi -2 c k p c v -1 2 2 b s -1 b m z 2 b m -1 2 b s 1 2 -1 4 b m -1 2 b s 2 b s -1 b m -1 -1 -2 c k p c v 2 4 2 b s -1 b m -1 2 b s -1 b m 1 2 b m -1 2 b s Erfi -2 c k p c v -1 2 b m -1 2 b s z 2 2 b s -1 b m 1 2 -1 -1 2 c k p -1 c v 2 4 b m -1 2 b s -1 2 b s -1 b m b m -1 2 b s 1 2 Erfi 2 c k p -1 c v 2 b m -1 2 b s z 2 b m -1 2 b s 1 2 -1 4 b m -1 2 b s 2 b s -1 b m -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