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 Exp

 http://functions.wolfram.com/01.03.21.0639.01

 Input Form

 Integrate[z^n E^(b z^2) (E^(f z + g))^\[Nu], z] == (-(1/(2 Sqrt[b]))) (((E^(g + f z))^\[Nu] Sum[2^(-n + q) b^(-(1/2) - n) ((-f) \[Nu])^(n - q) (2 b z + f \[Nu])^(1 + q) (-((2 b z + f \[Nu])^2/b))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((2 b z + f \[Nu])^2/(4 b))], {q, 0, n}])/E^((f \[Nu] (4 b z + f \[Nu]))/(4 b))) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox["\[ExponentialE]", RowBox[List["b", " ", SuperscriptBox["z", "2"]]]], SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["f", " ", "z"]], "+", "g"]]], ")"]], "\[Nu]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", " ", SqrtBox["b"]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[RowBox[List["f", " ", "\[Nu]", " ", RowBox[List["(", RowBox[List[RowBox[List["4", " ", "b", " ", "z"]], "+", RowBox[List["f", " ", "\[Nu]"]]]], ")"]]]], RowBox[List["4", " ", "b"]]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["g", "+", RowBox[List["f", " ", "z"]]]]], ")"]], "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox["b", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "f"]], " ", "\[Nu]"]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List["f", " ", "\[Nu]"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List["f", " ", "\[Nu]"]]]], ")"]], "2"], "b"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "b", " ", "z"]], "+", RowBox[List["f", " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["4", " ", "b"]]]]]]], "]"]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n b z 2 ( g + f z ) ν z - 1 2 b - f ν ( 4 b z + f ν ) 4 b ( g + f z ) ν q = 0 n 2 q - n b - n - 1 2 ( - f ν ) n - q ( 2 b z + f ν ) q + 1 ( - ( 2 b z + f ν ) 2 b ) 1 2 ( - q - 1 ) ( n q ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["q", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] Γ ( q + 1 2 , - ( 2 b z + f ν ) 2 4 b ) /; n Condition z z n b z 2 g f z ν -1 1 2 b 1 2 -1 -1 f ν 4 b z f ν 4 b -1 g f z ν q 0 n 2 q -1 n b -1 n -1 1 2 -1 f ν n -1 q 2 b z f ν q 1 -1 2 b z f ν 2 b -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 2 b z f ν 2 4 b -1 n [/itex]

 Rule Form

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

 2002-12-18