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/01.20.21.3845.01

 Input Form

 Integrate[E^(p z) Cos[b z]^m Cosh[c Sqrt[z]]^v, z] == (1/p) ((-1)^v 2^(-m - v) E^(p z) Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2])) + (-1)^v 2^(-m - v) Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(E^(((-I) b (-2 k + m) + p) z)/((-I) b (-2 k + m) + p) + E^((I b (-2 k + m) + p) z)/(I b (-2 k + m) + p)) Binomial[m, k], {k, 0, Floor[(1/2) (-1 + m)]}] + 2^(-1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[Binomial[v, s] ((4 E^(p z) Cosh[c (-2 s + v) Sqrt[z]])/p - (1/p^(3/2)) ((c Sqrt[Pi] (-2 s + v) (Erfi[(c (-2 s + v) - 2 p Sqrt[z])/ (2 Sqrt[p])] + Erfi[(c (-2 s + v) + 2 p Sqrt[z])/(2 Sqrt[p])]))/ E^((c^2 (-2 s + v)^2)/(4 p)))), {s, 0, Floor[(1/2) (-1 + v)]}] + 2^(-1 - m - v) Sum[Binomial[v, s] Sum[Binomial[m, k] ((2 E^((-c) (-2 s + v) Sqrt[z] + ((-I) b (2 k - m) + p) z))/ ((-I) b (2 k - m) + p) + (2 E^(c (-2 s + v) Sqrt[z] + (I b (2 k - m) + p) z))/(I b (2 k - m) + p) + (2 E^((-c) (-2 s + v) Sqrt[z] + ((-I) b (-2 k + m) + p) z))/ ((-I) b (-2 k + m) + p) + (2 E^(c (-2 s + v) Sqrt[z] + (I b (-2 k + m) + p) z))/ (I b (-2 k + m) + p) - (c Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 (I b (2 k - m) - p) Sqrt[z])/ (2 Sqrt[(-I) b (2 k - m) + p])])/E^((c^2 (-2 s + v)^2)/ (4 ((-I) b (2 k - m) + p)))/((-I) b (2 k - m) + p)^(3/2) - (c Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 (I b (-2 k + m) - p) Sqrt[z])/(2 Sqrt[(-I) b (-2 k + m) + p])])/ E^((c^2 (-2 s + v)^2)/(4 ((-I) b (-2 k + m) + p)))/ ((-I) b (-2 k + m) + p)^(3/2) - (c Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 (I b (2 k - m) + p) Sqrt[z])/(2 Sqrt[I b (2 k - m) + p])])/ E^((c^2 (-2 s + v)^2)/(4 (I b (2 k - m) + p)))/ (I b (2 k - m) + p)^(3/2) - (c Sqrt[Pi] (-2 s + v) Erfi[(c (-2 s + v) + 2 (I b (-2 k + m) + p) Sqrt[z])/ (2 Sqrt[I b (-2 k + m) + p])])/E^((c^2 (-2 s + v)^2)/ (4 (I b (-2 k + m) + p)))/(I b (-2 k + m) + p)^(3/2)), {k, 0, Floor[(1/2) (-1 + m)]}], {s, 0, Floor[(1/2) (-1 + v)]}] /; Element[m, Integers] && m > 0 && Element[v, Integers] && v > 0

 Standard Form

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 MathML Form

 p z cos m ( b z ) cosh v ( c z ) z ( - 1 ) v 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]]]]] ( k = 0 m - 1 2 ( ( b ( m - 2 k ) + p ) z b ( m - 2 k ) + p + ( p - b ( m - 2 k ) ) z p - b ( m - 2 k ) ) ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ) ( 1 - v mod 2 \$CellContext`v 2 ) + ( - 1 ) v 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 + 2 - m - v - 1 ( 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]]]]] ( 4 p z cosh ( c ( v - 2 s ) z ) p - c - c 2 ( v - 2 s ) 2 4 p π ( v - 2 s ) ( erfi ( 2 z p + c ( v - 2 s ) 2 p ) + erfi ( c ( v - 2 s ) - 2 p z 2 p ) ) p 3 / 2 ) + 2 - m - v - 1 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]]]]] k = 0 m - 1 2 ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( 2 ( p - b ( 2 k - m ) ) z - c ( v - 2 s ) z p - b ( 2 k - m ) - c - c 2 ( v - 2 s ) 2 4 ( p - b ( 2 k - m ) ) π ( v - 2 s ) erfi ( 2 z ( b ( 2 k - m ) - p ) + c ( v - 2 s ) 2 p - b ( 2 k - m ) ) ( p - b ( 2 k - m ) ) 3 / 2 + 2 c z ( v - 2 s ) + ( b ( 2 k - m ) + p ) z b ( 2 k - m ) + p - c - c 2 ( v - 2 s ) 2 4 ( b ( 2 k - m ) + p ) π ( v - 2 s ) erfi ( 2 z ( b ( 2 k - m ) + p ) + c ( v - 2 s ) 2 b ( 2 k - m ) + p ) ( b ( 2 k - m ) + p ) 3 / 2 + 2 ( p - b ( m - 2 k ) ) z - c ( v - 2 s ) z p - b ( m - 2 k ) - c - c 2 ( v - 2 s ) 2 4 ( p - b ( m - 2 k ) ) π ( v - 2 s ) erfi ( 2 z ( b ( m - 2 k ) - p ) + c ( v - 2 s ) 2 p - b ( m - 2 k ) ) ( p - b ( m - 2 k ) ) 3 / 2 + 2 c z ( v - 2 s ) + ( b ( m - 2 k ) + p ) z b ( m - 2 k ) + p - c - c 2 ( v - 2 s ) 2 4 ( b ( m - 2 k ) + p ) π ( v - 2 s ) erfi ( 2 z ( b ( m - 2 k ) + p ) + c ( v - 2 s ) 2 b ( m - 2 k ) + p ) ( b ( m - 2 k ) + p ) 3 / 2 ) /; m + v + Condition z p z b z m c z 1 2 v -1 v 2 -1 m -1 v Binomial v v 2 -1 k 0 m -1 2 -1 b m -1 2 k p z b m -1 2 k p -1 p -1 b m -1 2 k z p -1 b m -1 2 k -1 Binomial m k 1 -1 \$CellContext`v 2 -1 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 2 -1 m -1 v -1 Binomial m m 2 -1 1 -1 \$CellContext`m 2 s 0 v -1 2 -1 Binomial v s 4 p z c v -1 2 s z 1 2 p -1 -1 c -1 c 2 v -1 2 s 2 4 p -1 1 2 v -1 2 s Erfi 2 z 1 2 p c v -1 2 s 2 p 1 2 -1 Erfi c v -1 2 s -1 2 p z 1 2 2 p 1 2 -1 p 3 2 -1 2 -1 m -1 v -1 s 0 v -1 2 -1 Binomial v s k 0 m -1 2 -1 Binomial m k 2 p -1 b 2 k -1 m z -1 c v -1 2 s z 1 2 p -1 b 2 k -1 m -1 -1 c -1 c 2 v -1 2 s 2 4 p -1 b 2 k -1 m -1 1 2 v -1 2 s Erfi 2 z 1 2 b 2 k -1 m -1 p c v -1 2 s 2 p -1 b 2 k -1 m 1 2 -1 p -1 b 2 k -1 m 3 2 -1 2 c z 1 2 v -1 2 s b 2 k -1 m p z b 2 k -1 m p -1 -1 c -1 c 2 v -1 2 s 2 4 b 2 k -1 m p -1 1 2 v -1 2 s Erfi 2 z 1 2 b 2 k -1 m p c v -1 2 s 2 b 2 k -1 m p 1 2 -1 b 2 k -1 m p 3 2 -1 2 p -1 b m -1 2 k z -1 c v -1 2 s z 1 2 p -1 b m -1 2 k -1 -1 c -1 c 2 v -1 2 s 2 4 p -1 b m -1 2 k -1 1 2 v -1 2 s Erfi 2 z 1 2 b m -1 2 k -1 p c v -1 2 s 2 p -1 b m -1 2 k 1 2 -1 p -1 b m -1 2 k 3 2 -1 2 c z 1 2 v -1 2 s b m -1 2 k p z b m -1 2 k p -1 -1 c -1 c 2 v -1 2 s 2 4 b m -1 2 k p -1 1 2 v -1 2 s Erfi 2 z 1 2 b m -1 2 k p c v -1 2 s 2 b m -1 2 k p 1 2 -1 b m -1 2 k p 3 2 -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18