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 Exp

 http://functions.wolfram.com/01.03.21.0699.01

 Input Form

 Integrate[z^n E^(d z) (E^(c z^2 + f z + g))^\[Nu], z] == (-(1/(2 Sqrt[c \[Nu]]))) (E^((-z) (f + c z) \[Nu] - (d + f \[Nu])^2/(4 c \[Nu])) (E^(g + z (f + c z)))^\[Nu] Sum[2^(-n + q) (c \[Nu])^(-(1/2) - n) (-d - f \[Nu])^(n - q) (d + f \[Nu] + 2 c z \[Nu])^(1 + q) (-((d + f \[Nu] + 2 c z \[Nu])^2/(c \[Nu])))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((d + f \[Nu] + 2 c z \[Nu])^2/ (4 c \[Nu]))], {q, 0, n}]) /; 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["d", " ", "z"]]], SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["f", " ", "z"]], "+", "g"]]], ")"]], "\[Nu]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", " ", SqrtBox[RowBox[List["c", " ", "\[Nu]"]]]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["c", " ", "z"]]]], ")"]], " ", "\[Nu]"]], "-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", RowBox[List["f", " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["4", " ", "c", " ", "\[Nu]"]]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["g", "+", RowBox[List["z", " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]]]], ")"]], "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["c", " ", "\[Nu]"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "d"]], "-", RowBox[List["f", " ", "\[Nu]"]]]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", RowBox[List["f", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "c", " ", "z", " ", "\[Nu]"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["d", "+", RowBox[List["f", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "c", " ", "z", " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["c", " ", "\[Nu]"]]]]], ")"]], 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["d", "+", RowBox[List["f", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "c", " ", "z", " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["4", " ", "c", " ", "\[Nu]"]]]]]]], "]"]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

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

 Rule Form

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

 2002-12-18