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/01.20.21.4653.01

 Input Form

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