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 Exp

 http://functions.wolfram.com/01.03.21.0544.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

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

 2002-12-18