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 Exp

 http://functions.wolfram.com/01.03.21.0538.01

 Input Form

 Integrate[z^(n + 1/2) (E^(a z + b))^\[Nu], z] == (-E^((-a) z \[Nu])) (E^(b + a z))^\[Nu] Sqrt[z] (((-a) \[Nu])^(-1 - n)/Sqrt[(-a) z \[Nu]]) (Erfc[Sqrt[(-a) z \[Nu]]] Gamma[3/2 + n] + E^(a z \[Nu]) Sum[((-a) z \[Nu])^(1/2 + k)/Pochhammer[3/2 + n, k - n], {k, 0, n}] - E^(a z \[Nu]) Sum[((-a) z \[Nu])^(1/2 + k)/ Pochhammer[3/2 + n, k - n], {k, 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[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]"], " ", SqrtBox["z"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", "\[Nu]"]], ")"]], RowBox[List[RowBox[List["-", "1"]], "-", "n"]]], "/", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", "z", " ", "\[Nu]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Erfc", "[", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", "z", " ", "\[Nu]"]]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["3", "2"], "+", "n"]], "]"]]]], "+", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", "z", " ", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", "z", " ", "\[Nu]"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["a", " ", "z", " ", "\[Nu]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", RowBox[List["1", "+", "n"]]]], RowBox[List["-", "1"]]], FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "a"]], " ", "z", " ", "\[Nu]"]], ")"]], RowBox[List[FractionBox["1", "2"], "+", "k"]]], RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List[FractionBox["3", "2"], "+", "n"]], ",", RowBox[List["k", "-", "n"]]]], "]"]]]]]]]]], ")"]]]]]], "/;", RowBox[List["n", "\[Element]", "Integers"]]]]]]

 MathML Form

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

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

 2002-12-18