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 Exp

 http://functions.wolfram.com/01.03.21.0663.01

 Input Form

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

 MathML Form

 z n b z 2 + d z ( c z 2 ) ν z - 1 2 b + c ν ( - d 2 4 ( b + c ν ) - c z 2 ν ( c z 2 ) ν q = 0 n 2 q - n ( - d ) n - q ( b + c ν ) - n - 1 2 ( d + 2 z ( b + c ν ) ) q + 1 ( - ( d + 2 z ( b + c ν ) ) 2 b + 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 + 2 z ( b + c ν ) ) 2 4 ( b + c ν ) ) ) /; n Condition z z n b z 2 d z c z 2 ν -1 1 2 b c ν 1 2 -1 -1 d 2 4 b c ν -1 -1 c z 2 ν c z 2 ν q 0 n 2 q -1 n -1 d n -1 q b c ν -1 n -1 1 2 d 2 z b c ν q 1 -1 d 2 z b c ν 2 b c ν -1 1 2 -1 q -1 Binomial n q Gamma q 1 2 -1 -1 d 2 z b c ν 2 4 b c ν -1 n [/itex]

 Rule Form

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

 2002-12-18