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/01.20.21.4159.01

 Input Form

 Integrate[Sinh[b z^2 + e]^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 E^(2 e k - e m) Binomial[m, k] ((-1)^m Sqrt[b (2 k - m)] Erfi[Sqrt[b (2 k - m)] z] + E^(2 e (-2 k + m)) Sqrt[b (-2 k + m)] Erfi[(b (2 k - m) z)/ Sqrt[b (-2 k + m)]])), {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^(e (2 k - m) + g (2 s - v) - (-2 f s + f v)^2/(8 b k - 4 b m)) Sqrt[2 b k - b m] Erfi[(f (2 s - v) + 4 b k z - 2 b m z)/(2 Sqrt[2 b k - b m])])) + (E^(e (-2 k + m) + g (-2 s + v) + (-2 f s + f v)^2/(8 b k - 4 b m)) Erfi[((-f) (2 s - v) - 4 b k z + 2 b m z)/(2 Sqrt[b (-2 k + m)])])/ Sqrt[b (-2 k + m)] + (1/(-2 b k + b m)) (E^((-e) (2 k + m) + g (-2 s + v) - (f^2 (-2 s + v)^2)/ (8 b k - 4 b m)) ((-(-1)^m) E^(4 e k) Sqrt[2 b k - b m] Erfi[((-f) (2 s - v) + 4 b k z - 2 b m z)/(2 Sqrt[2 b k - b m])] + E^(2 (e m + g (2 s - v) + (-2 f s + f v)^2/ (8 b k - 4 b m))) Sqrt[-2 b k + b m] Erfi[(f (2 s - v) - 4 b k z + 2 b m 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

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "+", "e"]], "]"]], "m"], SuperscriptBox[RowBox[List["Cosh", "[", RowBox[List[RowBox[List["f", " ", "z"]], "+", "g"]], "]"]], "v"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List[SuperscriptBox["\[ImaginaryI]", "m"], " ", SuperscriptBox["2", RowBox[List[RowBox[List["-", "m"]], "-", "v"]]], " ", "z", " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["v", ",", FractionBox["v", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["v", ",", "2"]], "]"]]]], ")"]]]], "-", FractionBox["1", "b"], RowBox[List["(", RowBox[List[SuperscriptBox["2", 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"0"]]]]]]]]

 MathML Form

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