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 Exp

 http://functions.wolfram.com/01.03.21.0542.01

 Input Form

 Integrate[z^(2 n) (E^(a z^2))^\[Nu], z] == ((-(1/2)) (E^(a z^2))^\[Nu] z^(1 + 2 n) ((-a) z^2 \[Nu])^(-(1/2) - n) (Erfc[Sqrt[(-a) z^2 \[Nu]]] Gamma[1/2 + n] + E^(a z^2 \[Nu]) Sum[((-a) z^2 \[Nu])^(1/2 + j)/Pochhammer[1/2 + n, 1 + j - n], {j, 0, -1 + n}] - E^(a z^2 \[Nu]) Sum[((-a) z^2 \[Nu])^(1/2 + j)/Pochhammer[1/2 + n, 1 + j - n], {j, n, -1}]))/E^(a z^2 \[Nu]) /; Element[n, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["2", "n"]]], SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]], ")"]], "\[Nu]"], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]], ")"]], "\[Nu]"], " ", SuperscriptBox["z", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Erfc", "[", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "n"]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "j"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "n"]], ",", RowBox[List["1", "+", "j", "-", "n"]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "n"]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "j"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "n"]], ",", RowBox[List["1", "+", "j", "-", "n"]]]], "]"]]]]]]]]], ")"]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]

 MathML Form

 z 2 n ( a z 2 ) ν z - 1 2 - a z 2 ν ( a z 2 ) ν z 2 n + 1 ( - a z 2 ν ) - n - 1 2 ( erfc ( - a z 2 ν ) Γ ( n + 1 2 ) + a z 2 ν j = 0 n - 1 ( - a z 2 ν ) j + 1 2 ( n + 1 2 ) j - n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["j", "-", "n", "+", "1"]]], Pochhammer] - a z 2 ν j = n - 1 ( - a z 2 ν ) j + 1 2 ( n + 1 2 ) j - n + 1 TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["1", "2"]]], ")"]], RowBox[List["j", "-", "n", "+", "1"]]], Pochhammer] ) /; n TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition z z 2 n a z 2 ν -1 1 2 -1 a z 2 ν a z 2 ν z 2 n 1 -1 a z 2 ν -1 n -1 1 2 Erfc -1 a z 2 ν 1 2 Gamma n 1 2 a z 2 ν j 0 n -1 -1 a z 2 ν j 1 2 Pochhammer n 1 2 j -1 n 1 -1 -1 a z 2 ν j n -1 -1 a z 2 ν j 1 2 Pochhammer n 1 2 j -1 n 1 -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["2", " ", "n_"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["a_", " ", SuperscriptBox["z_", "2"]]]], ")"]], "\[Nu]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SuperscriptBox["z", "2"]]]], ")"]], "\[Nu]"], " ", SuperscriptBox["z", RowBox[List["1", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Erfc", "[", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["1", "2"], "+", "n"]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], RowBox[List[RowBox[List["-", "1"]], "+", "n"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "j"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "n"]], ",", RowBox[List["1", "+", "j", "-", "n"]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "n"]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "j"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], "+", "n"]], ",", RowBox[List["1", "+", "j", "-", "n"]]]], "]"]]]]]]]]], ")"]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]]]

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

 2002-12-18