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 Cosh

 http://functions.wolfram.com/01.20.21.1560.01

 Input Form

 Integrate[Sinh[b z^2]^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 - f^2/(8 b k - 4 b m)) Sqrt[2 b k - b m] Erfi[(-f + 4 b k z - 2 b m z)/ (2 Sqrt[2 b k - b m])])) + (1/(-2 b k + b m)) (E^(g - f^2/(8 b k - 4 b m)) ((-(-1)^m) Sqrt[2 b k - b m] Erfi[(f + 4 b k z - 2 b m z)/(2 Sqrt[2 b k - b m])] + E^(2 (-g + f^2/(8 b k - 4 b m))) Sqrt[-2 b k + b m] Erfi[(-f - 4 b k z + 2 b m z)/(2 Sqrt[-2 b k + b m])])) + (E^(g + f^2/(8 b k - 4 b m)) Erfi[(f - 4 b k z + 2 b m 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["b", " ", SuperscriptBox["z", "2"]]], "]"]], "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"]], "-", FractionBox[SuperscriptBox["f", "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["4", " ", "b", " ", "k", " ", "z"]], "-", RowBox[List["2", " ", "b", " ", "m", " ", "z"]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]]]]], "]"]]]], ")"]]]]]], "+", FractionBox["1", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", "b", " ", "k"]], "+", RowBox[List["b", " ", "m"]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["g", "-", FractionBox[SuperscriptBox["f", "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"]]], " ", SqrtBox[RowBox[List[RowBox[List["2", " ", "b", " ", "k"]], "-", RowBox[List["b", " ", "m"]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["f", "+", RowBox[List["4", " ", "b", " ", "k", " ", "z"]], "-", RowBox[List["2", " ", "b", " ", "m", " ", "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"]], "+", FractionBox[SuperscriptBox["f", "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["4", " ", "b", " ", "k", " ", "z"]], "+", RowBox[List["2", " ", "b", " ", "m", " ", "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", "+", FractionBox[SuperscriptBox["f", "2"], RowBox[List[RowBox[List["8", " ", "b", " ", "k"]], "-", RowBox[List["4", " ", "b", " ", "m"]]]]]]]], " ", RowBox[List["Erfi", "[", FractionBox[RowBox[List["f", "-", RowBox[List["4", " ", "b", " ", "k", " ", "z"]], "+", RowBox[List["2", " ", "b", " ", "m", " ", "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 ) 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 8 b k - 4 b m + g erfi ( f - 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 8 b k - 4 b m + g ( 2 ( f 2 8 b k - 4 b m - g ) b m - 2 b k erfi ( - f - 4 b k z + 2 b m z 2 b m - 2 b k ) - ( - 1 ) m 2 b k - b m erfi ( f + 4 b k z - 2 b m z 2 2 b k - b m ) ) ) - ( - 1 ) m - f 2 8 b k - 4 b m - g 2 b k - b m erfi ( - f + 4 b k z - 2 b m z 2 2 b k - b m ) b ( m - 2 k ) ) /; m + Condition z b z 2 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 2 8 b k -1 4 b m -1 g Erfi f -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 8 b k -1 4 b m -1 g 2 f 2 8 b k -1 4 b m -1 -1 g b m -1 2 b k 1 2 Erfi -1 f -1 4 b k z 2 b m z 2 b m -1 2 b k 1 2 -1 -1 -1 m 2 b k -1 b m 1 2 Erfi f 4 b k z -1 2 b m z 2 2 b k -1 b m 1 2 -1 -1 -1 m -1 f 2 8 b k -1 4 b m -1 -1 g 2 b k -1 b m 1 2 Erfi -1 f 4 b k z -1 2 b m z 2 2 b k -1 b m 1 2 -1 b m -1 2 k -1 m SuperPlus [/itex]

 Rule Form

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

 2002-12-18