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 Exp

 http://functions.wolfram.com/01.03.21.0626.01

 Input Form

 Integrate[z^(n + 1/2) E^(c z) (E^(a z + b))^\[Nu], z] == (-E^((-a) z \[Nu])) (E^(b + a z))^\[Nu] z^(3/2 + n) ((-z) (c + a \[Nu]))^(-(3/2) - n) (Erfc[Sqrt[(-z) (c + a \[Nu])]] Gamma[3/2 + n] + E^(z (c + a \[Nu])) Sum[((-z) (c + a \[Nu]))^(1/2 + j)/Pochhammer[3/2 + n, j - n], {j, 0, n}] - E^(z (c + a \[Nu])) Sum[((-z) (c + a \[Nu]))^(1/2 + j)/ Pochhammer[3/2 + n, j - n], {j, 1 + n, -1}]) /; Element[n, Integers]

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", RowBox[List["n", "+", FractionBox["1", "2"]]]], SuperscriptBox["\[ExponentialE]", RowBox[List["c", " ", "z"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["a", " ", "z"]], "+", "b"]]], ")"]], "\[Nu]"], " ", RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", "z", " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]]], ")"]], "\[Nu]"], " ", SuperscriptBox["z", RowBox[List[FractionBox["3", "2"], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Erfc", "[", SqrtBox[RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "j"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["j", "-", "n"]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", RowBox[List["1", "+", "n"]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "j"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["j", "-", "n"]]]], "]"]]]]]]]]], ")"]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]

 MathML Form

 z n + 1 2 c z ( a z + b ) ν z - - a z ν ( b + a z ) ν z n + 3 2 ( - z ( c + a ν ) ) - n - 3 2 ( erfc ( - z ( c + a ν ) ) Γ ( n + 3 2 ) + z ( c + a ν ) j = 0 n ( - z ( c + a ν ) ) j + 1 2 ( n + 3 2 ) j - n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["3", "2"]]], ")"]], RowBox[List["j", "-", "n"]]], Pochhammer] - z ( c + a ν ) j = n + 1 - 1 ( - z ( c + a ν ) ) j + 1 2 ( n + 3 2 ) j - n TagBox[SubscriptBox[RowBox[List["(", RowBox[List["n", "+", FractionBox["3", "2"]]], ")"]], RowBox[List["j", "-", "n"]]], Pochhammer] ) /; n TagBox["\[DoubleStruckCapitalZ]", Function[Integers]] Condition z z n 1 2 c z a z b ν -1 -1 a z ν b a z ν z n 3 2 -1 z c a ν -1 n -1 3 2 Erfc -1 z c a ν 1 2 Gamma n 3 2 z c a ν j 0 n -1 z c a ν j 1 2 Pochhammer n 3 2 j -1 n -1 -1 z c a ν j n 1 -1 -1 z c a ν j 1 2 Pochhammer n 3 2 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["n_", "+", FractionBox["1", "2"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["c_", " ", "z_"]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["a_", " ", "z_"]], "+", "b_"]]], ")"]], "\[Nu]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["-", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "a"]], " ", "z", " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["b", "+", RowBox[List["a", " ", "z"]]]]], ")"]], "\[Nu]"], " ", SuperscriptBox["z", RowBox[List[FractionBox["3", "2"], "+", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["3", "2"]]], "-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Erfc", "[", SqrtBox[RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "j"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["j", "-", "n"]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", RowBox[List["1", "+", "n"]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "z"]], " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["a", " ", "\[Nu]"]]]], ")"]]]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "j"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["j", "-", "n"]]]], "]"]]]]]]]]], ")"]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]]]

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

 2002-12-18