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 Exp

 http://functions.wolfram.com/01.03.21.0543.01

 Input Form

 Integrate[z^(2 n + 1) (E^(a z^2))^\[Nu], z] == ((-(1/2)) (E^(a z^2))^\[Nu] ((-a) \[Nu])^(-1 - n) (((-1)^n ExpIntegralEi[a z^2 \[Nu]])/(-1 - n)! + E^(a z^2 \[Nu]) Sum[((-a) z^2 \[Nu])^j/Pochhammer[1 + n, j - n], {j, 0, n}] - E^(a z^2 \[Nu]) Sum[((-a) z^2 \[Nu])^j/ Pochhammer[1 + n, j - n], {j, 1 + 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[RowBox[List["2", "n"]], "+", "1"]]], 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[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", "\[Nu]"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["ExpIntegralEi", "[", RowBox[List["a", " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]], "]"]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "n"]], ")"]], "!"]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]], ")"]], "j"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["j", "-", "n"]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", RowBox[List["1", "+", "n"]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", SuperscriptBox["z", "2"], " ", "\[Nu]"]], ")"]], "j"], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["1", "+", "n"]], ",", RowBox[List["j", "-", "n"]]]], "]"]]]]]]]]], ")"]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]

 MathML Form

 z 2 n + 1 ( a z 2 ) ν z - 1 2 - a z 2 ν ( a z 2 ) ν ( - a ν ) - n - 1 ( ( - 1 ) n Ei ( a z 2 ν ) ( - n - 1 ) ! + a z 2 ν j = 0 n ( - a z 2 ν ) j ( n + 1 ) j - n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], RowBox[List["j", "-", "n"]]], Pochhammer] - a z 2 ν j = n + 1 - 1 ( - a z 2 ν ) j ( n + 1 ) j - n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], RowBox[List["j", "-", "n"]]], Pochhammer] ) /; n TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition z z 2 n 1 a z 2 ν -1 1 2 -1 a z 2 ν a z 2 ν -1 a ν -1 n -1 -1 n ExpIntegralEi a z 2 ν -1 n -1 -1 a z 2 ν j 0 n -1 a z 2 ν j Pochhammer n 1 j -1 n -1 -1 a z 2 ν j n 1 -1 -1 a z 2 ν j Pochhammer n 1 j -1 n -1 n [/itex]

 Rule Form

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

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

 2002-12-18