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/01.20.21.4161.01

 Input Form

 Integrate[Sinh[b z^2 + d z]^m Cosh[f z + g]^v, z] == I^m 2^(-m - v) z Binomial[m, m/2] Binomial[v, v/2] (1 - Mod[m, 2]) (1 - Mod[v, 2]) - (1/b) (2^(-1 - m - v) Sqrt[Pi] Binomial[v, v/2] (1 - Mod[v, 2]) Sum[(1/(-2 k + m)) (((-1)^k Binomial[m, k] ((-1)^m E^((d^2 (-2 k + m))/(2 b)) Sqrt[b (2 k - m)] Erfi[(Sqrt[b (2 k - m)] (d + 2 b z))/(2 b)] + Sqrt[b (-2 k + m)] Erfi[((2 k - m) (d + 2 b z))/(2 Sqrt[b (-2 k + m)])]))/ E^((d^2 (-2 k + m))/(4 b))), {k, 0, Floor[(1/2) (-1 + m)]}]) - (1/f) (I^m 2^(1 - m - v) Binomial[m, m/2] (1 - Mod[m, 2]) Sum[(Binomial[v, s] Sinh[(2 s - v) (g + f z)])/(-2 s + v), {s, 0, Floor[(1/2) (-1 + v)]}]) + 2^(-1 - m - v) Sqrt[Pi] Sum[(-1)^k Binomial[m, k] Sum[Binomial[v, s] (-((1/(b (-2 k + m))) ((-1)^m E^(g (2 s - v) - (-2 d k + d m - 2 f s + f v)^2/(8 b k - 4 b m)) Sqrt[2 b k - b m] Erfi[(f (2 s - v) + 2 k (d + 2 b z) - m (d + 2 b z))/ (2 Sqrt[2 b k - b m])])) + (1/Sqrt[b (-2 k + m)]) (E^(g (-2 s + v) + (-2 d k + d m - 2 f s + f v)^2/(8 b k - 4 b m)) Erfi[((-f) (2 s - v) - 2 k (d + 2 b z) + m (d + 2 b z))/ (2 Sqrt[b (-2 k + m)])]) + (1/(-2 b k + b m)) (E^(g (-2 s + v) - (2 d k - d m + f (-2 s + v))^2/(8 b k - 4 b m)) ((-(-1)^m) Sqrt[2 b k - b m] Erfi[((-f) (2 s - v) + 2 k (d + 2 b z) - m (d + 2 b z))/(2 Sqrt[2 b k - b m])] + E^(2 (g (2 s - v) + (2 d k - d m - 2 f s + f v)^2/(8 b k - 4 b m))) Sqrt[-2 b k + b m] Erfi[(f (2 s - v) - 2 k (d + 2 b z) + m (d + 2 b z))/(2 Sqrt[-2 b k + b 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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RowBox[List["2", " ", "f", " ", "s"]], "+", RowBox[List["f", " ", "v"]]]], ")"]], "2"], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]]]]]]], ")"]]]]], " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]]], " ", RowBox[List["Erfi", "[", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["f", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "s"]], "-", "v"]], ")"]]]], "-", RowBox[List["2", " ", "k", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["2", " ", "b", " ", "z"]]]], ")"]]]], "+", RowBox[List["m", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["2", " ", "b", " ", "z"]]]], ")"]]]]]], ")"]], "/", RowBox[List["(", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]]]]], ")"]]]], "]"]]]]]], ")"]]]], ")"]]]], ")"]]]]]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "\[And]", RowBox[List["m", ">", "0"]], "\[And]", RowBox[List["v", "\[Element]", "Integers"]], "\[And]", RowBox[List["v", ">", "0"]]]]]]]]

 MathML Form

 sinh m ( b z 2 + d z ) cosh v ( g + f z ) z m 2 - m - v z ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["v", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - m mod 2 \$CellContext`m 2 ) ( 1 - v mod 2 \$CellContext`v 2 ) - 1 b ( 2 - m - v - 1 π ( v v 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["v", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - v mod 2 \$CellContext`v 2 ) k = 0 m - 1 2 1 m - 2 k ( ( - 1 ) k - d 2 ( m - 2 k ) 4 b ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( - 1 ) m d 2 ( m - 2 k ) 2 b b ( 2 k - m ) erfi ( b ( 2 k - m ) ( d + 2 b z ) 2 b ) + b ( m - 2 k ) erfi ( ( 2 k - m ) ( d + 2 b z ) 2 b ( m - 2 k ) ) ) ) ) - m 2 - m - v + 1 ( m m 2 ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox[FractionBox["m", "2"], Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( 1 - m mod 2 \$CellContext`m 2 ) s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] sinh ( ( 2 s - v ) ( g + f z ) ) v - 2 s f + 2 - m - v - 1 π k = 0 m - 1 2 ( - 1 ) k ( m k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["m", Identity, Rule[Editable, True]]], List[TagBox["k", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] s = 0 v - 1 2 ( v s ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["v", Identity, Rule[Editable, True]]], List[TagBox["s", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] ( ( - 2 d k + d m - 2 f s + f v ) 2 8 b k - 4 b m + g ( v - 2 s ) erfi ( - f ( 2 s - v ) - 2 k ( d + 2 b z ) + m ( d + 2 b z ) 2 b ( m - 2 k ) ) b ( m - 2 k ) + 1 b m - 2 b k ( g ( v - 2 s ) - ( 2 d k - d m + f ( v - 2 s ) ) 2 8 b k - 4 b m ( 2 ( ( 2 d k - d m - 2 f s + f v ) 2 8 b k - 4 b m + g ( 2 s - v ) ) b m - 2 b k erfi ( f ( 2 s - v ) - 2 k ( d + 2 b z ) + m ( d + 2 b z ) 2 b m - 2 b k ) - ( - 1 ) m 2 b k - b m erfi ( - f ( 2 s - v ) + 2 k ( d + 2 b z ) - m ( d + 2 b z ) 2 2 b k - b m ) ) ) - 1 b ( m - 2 k ) ( ( - 1 ) m g ( 2 s - v ) - ( - 2 d k + d m - 2 f s + f v ) 2 8 b k - 4 b m 2 b k - b m erfi ( f ( 2 s - v ) + 2 k ( d + 2 b z ) - m ( d + 2 b z ) 2 2 b k - b m ) ) ) /; m + v + Condition z b z 2 d z m g f z v m 2 -1 m -1 v z Binomial m m 2 -1 Binomial v v 2 -1 1 -1 \$CellContext`m 2 1 -1 \$CellContext`v 2 -1 1 b -1 2 -1 m -1 v -1 1 2 Binomial v v 2 -1 1 -1 \$CellContext`v 2 k 0 m -1 2 -1 1 m -1 2 k -1 -1 k -1 d 2 m -1 2 k 4 b -1 Binomial m k -1 m d 2 m -1 2 k 2 b -1 b 2 k -1 m 1 2 Erfi b 2 k -1 m 1 2 d 2 b z 2 b -1 b m -1 2 k 1 2 Erfi 2 k -1 m d 2 b z 2 b m -1 2 k 1 2 -1 -1 m 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 2 s -1 v g f z v -1 2 s -1 f -1 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 -2 d k d m -1 2 f s f v 2 8 b k -1 4 b m -1 g v -1 2 s Erfi -1 f 2 s -1 v -1 2 k d 2 b z m d 2 b z 2 b m -1 2 k 1 2 -1 b m -1 2 k 1 2 -1 1 b m -1 2 b k -1 g v -1 2 s -1 2 d k -1 d m f v -1 2 s 2 8 b k -1 4 b m -1 2 2 d k -1 d m -1 2 f s f v 2 8 b k -1 4 b m -1 g 2 s -1 v b m -1 2 b k 1 2 Erfi f 2 s -1 v -1 2 k d 2 b z m d 2 b z 2 b m -1 2 b k 1 2 -1 -1 -1 m 2 b k -1 b m 1 2 Erfi -1 f 2 s -1 v 2 k d 2 b z -1 m d 2 b z 2 2 b k -1 b m 1 2 -1 -1 1 b m -1 2 k -1 -1 m g 2 s -1 v -1 -2 d k d m -1 2 f s f v 2 8 b k -1 4 b m -1 2 b k -1 b m 1 2 Erfi f 2 s -1 v 2 k d 2 b z -1 m d 2 b z 2 2 b k -1 b m 1 2 -1 m SuperPlus v SuperPlus [/itex]

 Rule Form

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

 2002-12-18