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 Exp

 http://functions.wolfram.com/01.03.21.0640.01

 Input Form

 Integrate[z^n E^(b Sqrt[z]) (E^(f z + g))^\[Nu], z] == (2^(-1 - 2 n) E^(-(b^2/(4 f \[Nu])) - f z \[Nu]) (E^(g + f z))^\[Nu] Sum[(-1)^(-h + k) 4^k b^(-h - k + 2 n) (b + 2 f Sqrt[z] \[Nu])^(h + k) (-((b + 2 f Sqrt[z] \[Nu])^2/(f \[Nu])))^((1/2) (-1 - h - k)) Binomial[k, h] Binomial[n, k] (b (b + 2 f Sqrt[z] \[Nu]) Gamma[(1/2) (1 + h + k), -((b + 2 f Sqrt[z] \[Nu])^2/(4 f \[Nu]))] + 2 f \[Nu] Sqrt[-((b + 2 f Sqrt[z] \[Nu])^2/(f \[Nu]))] Gamma[(1/2) (2 + h + k), -((b + 2 f Sqrt[z] \[Nu])^2/(4 f \[Nu]))]), {k, 0, n}, {h, 0, k}])/(f \[Nu])^(2 (1 + n)) /; Element[n, Integers] && n >= 0

 Standard Form

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 MathML Form

 z n z b ( f z + g ) ν z 2 - 2 n - 1 - b 2 4 f ν - f z ν ( g + f z ) ν ( f ν ) - 2 ( n + 1 ) k = 0 n h = 0 k ( - 1 ) k - h 4 k b - h - k + 2 n ( b + 2 f z ν ) h + k ( - ( b + 2 f z ν ) 2 f ν ) 1 2 ( - h - k - 1 ) ( k h ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity]], List[TagBox["h", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( n k ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["k", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] ( b ( b + 2 f z ν ) Γ ( 1 2 ( h + k + 1 ) , - ( b + 2 f z ν ) 2 4 f ν ) + 2 f ν - ( b + 2 f z ν ) 2 f ν Γ ( 1 2 ( h + k + 2 ) , - ( b + 2 f z ν ) 2 4 f ν ) ) /; n Condition z z n z 1 2 b f z g ν 2 -2 n -1 -1 b 2 4 f ν -1 -1 f z ν g f z ν f ν -2 n 1 h 0 k k 0 n -1 k -1 h 4 k b -1 h -1 k 2 n b 2 f z 1 2 ν h k -1 b 2 f z 1 2 ν 2 f ν -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k b b 2 f z 1 2 ν Gamma 1 2 h k 1 -1 b 2 f z 1 2 ν 2 4 f ν -1 2 f ν -1 b 2 f z 1 2 ν 2 f ν -1 1 2 Gamma 1 2 h k 2 -1 b 2 f z 1 2 ν 2 4 f ν -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["b_", " ", SqrtBox["z_"]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["f_", " ", "z_"]], "+", "g_"]]], ")"]], "\[Nu]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "f", " ", "\[Nu]"]]]]], "-", RowBox[List["f", " ", "z", " ", "\[Nu]"]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["g", "+", RowBox[List["f", " ", "z"]]]]], ")"]], "\[Nu]"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["f", " ", "\[Nu]"]], ")"]], RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["h", "=", "0"]], "k"], RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "h"]], "+", "k"]]], " ", SuperscriptBox["4", "k"], " ", SuperscriptBox["b", RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", "\[Nu]"]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["f", " ", "\[Nu]"]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "h", "-", "k"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "h"]], "]"]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "k"]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", "\[Nu]"]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", "\[Nu]"]]]]]]], "]"]]]], "+", RowBox[List["2", " ", "f", " ", "\[Nu]", " ", SqrtBox[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["f", " ", "\[Nu]"]]]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "f", " ", SqrtBox["z"], " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["4", " ", "f", " ", "\[Nu]"]]]]]]], "]"]]]]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2002-12-18