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/01.20.21.1566.01

 Input Form

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

 MathML Form

 sinh m ( b z 2 + d z + e ) cosh ( g + f z ) 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 ) sinh ( g + f z ) ( 2 ) m f + 2 - m - 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]]]]] ( ( f - 2 d k + d m ) 2 8 b k - 4 b m + g + e ( m - 2 k ) erfi ( f - 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 ( - ( f + 2 d k - d m ) 2 8 b k - 4 b m + g - e ( 2 k + m ) ( 2 ( ( f + 2 d k - d m ) 2 8 b k - 4 b m - g + e m ) b m - 2 b k erfi ( - f - 2 k ( d + 2 b z ) + m ( d + 2 b z ) 2 b m - 2 b k ) - ( - 1 ) m 4 e k 2 b k - b m erfi ( f + 2 k ( d + 2 b z ) - m ( d + 2 b z ) 2 2 b k - b m ) ) ) - ( - 1 ) m - ( f - 2 d k + d m ) 2 8 b k - 4 b m - g + e ( 2 k - m ) 2 b k - b m erfi ( - f + 2 k ( d + 2 b z ) - m ( d + 2 b z ) 2 2 b k - b m ) b ( m - 2 k ) ) /; m + Condition z b z 2 d z e m g f z Binomial m m 2 -1 1 -1 \$CellContext`m 2 g f z 2 -1 m f -1 2 -1 m -2 1 2 k 0 m -1 2 -1 -1 k Binomial m k f -1 2 d k d m 2 8 b k -1 4 b m -1 g e m -1 2 k Erfi f -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 -1 f 2 d k -1 d m 2 8 b k -1 4 b m -1 g -1 e 2 k m 2 f 2 d k -1 d m 2 8 b k -1 4 b m -1 -1 g e m b m -1 2 b k 1 2 Erfi -1 f -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 4 e k 2 b k -1 b m 1 2 Erfi f 2 k d 2 b z -1 m d 2 b z 2 2 b k -1 b m 1 2 -1 -1 -1 m -1 f -1 2 d k d m 2 8 b k -1 4 b m -1 -1 g e 2 k -1 m 2 b k -1 b m 1 2 Erfi -1 f 2 k d 2 b z -1 m d 2 b z 2 2 b k -1 b m 1 2 -1 b m -1 2 k -1 m SuperPlus [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox[RowBox[List["Sinh", "[", RowBox[List[RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["d_", " ", "z_"]], "+", "e_"]], "]"]], "m_"], " ", RowBox[List["Cosh", "[", RowBox[List[RowBox[List["f_", " ", "z_"]], "+", "g_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", FractionBox["\[ImaginaryI]", "2"], ")"]], "m"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", FractionBox["m", "2"]]], "]"]], " ", RowBox[List["(", RowBox[List["1", "-", RowBox[List["Mod", "[", RowBox[List["m", ",", "2"]], "]"]]]], ")"]], " ", RowBox[List["Sinh", "[", RowBox[List["g", "+", RowBox[List["f", " ", "z"]]]], "]"]]]], "f"], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "2"]], "-", "m"]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["Floor", "[", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "m"]], ")"]]]], "]"]]], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["Binomial", "[", RowBox[List["m", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "g"]], "+", RowBox[List["e", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "m"]], ")"]]]], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["f", "-", RowBox[List["2", " ", "d", " ", "k"]], "+", RowBox[List["d", " ", "m"]]]], ")"]], "2"], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]]]]]]]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List[RowBox[List["-", "f"]], "+", 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["2", " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]]]]], "]"]]]], RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["g", "-", RowBox[List["e", " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "m"]], ")"]]]], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], "2"], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]]]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "m"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["4", " ", "e", " ", "k"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["f", "+", 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["2", " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]]]]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "g"]], "+", RowBox[List["e", " ", "m"]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["f", "+", RowBox[List["2", " ", "d", " ", "k"]], "-", RowBox[List["d", " ", "m"]]]], ")"]], "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", "[", FractionBox[RowBox[List[RowBox[List["-", "f"]], "-", 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["2", " ", SqrtBox[RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]]]]]], "]"]]]]]], ")"]]]], RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["g", "+", RowBox[List["e", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]], "+", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["f", "-", RowBox[List["2", " ", "d", " ", "k"]], "+", RowBox[List["d", " ", "m"]]]], ")"]], "2"], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["f", "-", 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["2", " ", SqrtBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]]], "]"]]]], SqrtBox[RowBox[List["b", " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "k"]], "+", "m"]], ")"]]]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", ">", "0"]]]]]]]]]]

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

 2002-12-18