Exponential function: Integration (formula 01.03.21.0211)

Exp

$Mathematica Notation$

Elementary Functions

Exp[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Arguments involving polynomials or algebraic functions and power factors

Involving power

Involving zⁿ d^{a z^r+b z+c}

http://functions.wolfram.com/01.03.21.0211.01

Input Form

Integrate[z^n d^(a z^2 + b z + c), z] == (-(1/(2 Sqrt[a Log[d]]))) (d^(-(b^2/(4 a)) + c) Sum[2^(-n + q) Binomial[n, q] Gamma[(1 + q)/2, -(((b + 2 a z)^2 Log[d])/(4 a))] (a Log[d])^(-(1/2) - n) ((-b) Log[d])^(n - q) ((b + 2 a z) Log[d])^ (1 + q) (-(((b + 2 a z)^2 Log[d])/a))^((1/2) (-1 - q)), {q, 0, n}]) /; Element[n, Integers] && n >= 0

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["\[Integral]", RowBox[List[SuperscriptBox["z", "n"], SuperscriptBox["d", RowBox[List[RowBox[List["a", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["b", " ", "z"]], "+", "c"]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["Log", "[", "d", "]"]]]]]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["d", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "a"]]]]], "+", "c"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]], "2"], " ", RowBox[List["Log", "[", "d", "]"]]]], RowBox[List["4", " ", "a"]]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", RowBox[List["Log", "[", "d", "]"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["Log", "[", "d", "]"]]]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]], " ", RowBox[List["Log", "[", "d", "]"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]], "2"], " ", RowBox[List["Log", "[", "d", "]"]]]], "a"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]]]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mo> ∫ </mo> <mrow> <mrow> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> ⁢ </mo> <msup> <mi> d </mi> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mi> c </mi> </mrow> </msup> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> d </mi> <mrow> <mi> c </mi> <mo> - </mo> <mfrac> <msup> <mi> b </mi> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> q </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msup> <mn> 2 </mn> <mrow> <mi> q </mi> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ⁢ </mo> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> q </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity]], List[TagBox["q", Identity]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] </annotation> </semantics> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mn> 2 </mn> </mfrac> <mo> , </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> a </mi> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> b </mi> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> q </mi> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mi> b </mi> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> a </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> <mi> a </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> q </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <ci> z </ci> <ci> n </ci> </apply> <apply> <power /> <ci> d </ci> <apply> <plus /> <apply> <times /> <ci> a </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> <ci> c </ci> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <ln /> <ci> d </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <ci> d </ci> <apply> <plus /> <ci> c </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> b </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> q </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> q </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> <apply> <ci> Binomial </ci> <ci> n </ci> <ci> q </ci> </apply> <apply> <ci> Gamma </ci> <apply> <times /> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ln /> <ci> d </ci> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> a </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <ci> a </ci> <apply> <ln /> <ci> d </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <ln /> <ci> d </ci> </apply> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> z </ci> </apply> </apply> <apply> <ln /> <ci> d </ci> </apply> </apply> <apply> <plus /> <ci> q </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> b </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> z </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ln /> <ci> d </ci> </apply> <apply> <power /> <ci> a </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", "n_"], " ", SuperscriptBox["d_", RowBox[List[RowBox[List["a_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["b_", " ", "z_"]], "+", "c_"]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["d", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["b", "2"], RowBox[List["4", " ", "a"]]]]], "+", "c"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]], "2"], " ", RowBox[List["Log", "[", "d", "]"]]]], RowBox[List["4", " ", "a"]]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["a", " ", RowBox[List["Log", "[", "d", "]"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["Log", "[", "d", "]"]]]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]], " ", RowBox[List["Log", "[", "d", "]"]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["b", "+", RowBox[List["2", " ", "a", " ", "z"]]]], ")"]], "2"], " ", RowBox[List["Log", "[", "d", "]"]]]], "a"]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]]]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List["a", " ", RowBox[List["Log", "[", "d", "]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

2002-12-18