html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 Exp

 http://functions.wolfram.com/01.03.21.0552.01

 Input Form

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

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["a", " ", SqrtBox["z"]]], "+", RowBox[List["b", " ", "z"]]]]], ")"]], "\[Nu]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Nu]"]], RowBox[List["4", " ", "b"]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["a", " ", SqrtBox["z"]]], "+", RowBox[List["b", " ", "z"]]]], ")"]], "\[Nu]"]]]]], " ", SuperscriptBox["b", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["a", " ", SqrtBox["z"]]], "+", RowBox[List["b", " ", "z"]]]]], ")"]], "\[Nu]"], 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["a", RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"], " ", "\[Nu]"]], "b"]]], ")"]], 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["a", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"], " ", "\[Nu]"]], RowBox[List["4", " ", "b"]]]]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["2", " ", "b"]], "\[Nu]"], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"], " ", "\[Nu]"]], "b"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"], " ", "\[Nu]"]], RowBox[List["4", " ", "b"]]]]]]], "]"]]]]]], ")"]]]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 z n ( z a + b z ) ν z 2 - 2 n - 1 - ν a 2 4 b - ( z a + b z ) ν b - 2 ( n + 1 ) ( z a + b z ) ν k = 0 n h = 0 k ( - 1 ) k - h 4 k a - h - k + 2 n ( a + 2 b z ) h + k ( - ( a + 2 b z ) 2 ν b ) 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]]]]] ( a ( a + 2 b z ) Γ ( 1 2 ( h + k + 1 ) , - ( a + 2 b z ) 2 ν 4 b ) + - ( a + 2 b z ) 2 ν b ( 2 b ) Γ ( 1 2 ( h + k + 2 ) , - ( a + 2 b z ) 2 ν 4 b ) ν ) /; n Condition z z n z 1 2 a b z ν 2 -2 n -1 -1 ν a 2 4 b -1 -1 z 1 2 a b z ν b -2 n 1 z 1 2 a b z ν h 0 k k 0 n -1 k -1 h 4 k a -1 h -1 k 2 n a 2 b z 1 2 h k -1 a 2 b z 1 2 2 ν b -1 1 2 -1 h -1 k -1 Binomial k h Binomial n k a a 2 b z 1 2 Gamma 1 2 h k 1 -1 a 2 b z 1 2 2 ν 4 b -1 -1 a 2 b z 1 2 2 ν b -1 1 2 2 b Gamma 1 2 h k 2 -1 a 2 b z 1 2 2 ν 4 b -1 ν -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[RowBox[List["a_", " ", SqrtBox["z_"]]], "+", RowBox[List["b_", " ", "z_"]]]]], ")"]], "\[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[RowBox[List[SuperscriptBox["a", "2"], " ", "\[Nu]"]], RowBox[List["4", " ", "b"]]]]], "-", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["a", " ", SqrtBox["z"]]], "+", RowBox[List["b", " ", "z"]]]], ")"]], " ", "\[Nu]"]]]]], " ", SuperscriptBox["b", RowBox[List[RowBox[List["-", "2"]], " ", RowBox[List["(", RowBox[List["1", "+", "n"]], ")"]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["a", " ", SqrtBox["z"]]], "+", RowBox[List["b", " ", "z"]]]]], ")"]], "\[Nu]"], " ", 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["a", RowBox[List[RowBox[List["-", "h"]], "-", "k", "+", RowBox[List["2", " ", "n"]]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], ")"]], RowBox[List["h", "+", "k"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"], " ", "\[Nu]"]], "b"]]], ")"]], 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["a", " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], ")"]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["1", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"], " ", "\[Nu]"]], RowBox[List["4", " ", "b"]]]]]]], "]"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["2", " ", "b"]], ")"]], " ", SqrtBox[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"], " ", "\[Nu]"]], "b"]]]], " ", RowBox[List["Gamma", "[", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["2", "+", "h", "+", "k"]], ")"]]]], ",", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["a", "+", RowBox[List["2", " ", "b", " ", SqrtBox["z"]]]]], ")"]], "2"], " ", "\[Nu]"]], RowBox[List["4", " ", "b"]]]]]]], "]"]]]], "\[Nu]"]]], ")"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2002-12-18