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 Exp

 http://functions.wolfram.com/01.03.21.0656.01

 Input Form

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

 Standard Form

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

 MathML Form

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

 Rule Form

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

 2002-12-18