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/01.20.21.4651.01

 Input Form

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