Exponential function: Integration (formula 01.03.21.0221)

Exp

$Mathematica Notation$

Elementary Functions

Exp[z]

Integration

Indefinite integration

Involving one direct function and elementary functions

Arguments involving rational functions and power factors

Involving power

Involving z²ⁿ d^{a z²+b/z²}

http://functions.wolfram.com/01.03.21.0221.01

Input Form

Integrate[z^(2 n) d^(a z^2 + b/z^2), z] == (Sqrt[Pi]/4) D[(1/(Log[d]^n Sqrt[(-a) Log[d]])) (Erfc[Sqrt[(-b) Log[d]]/z - Sqrt[(-a) Log[d]] z]/ E^(2 Sqrt[(-a) Log[d]] Sqrt[(-b) Log[d]]) - E^(2 Sqrt[(-a) Log[d]] Sqrt[(-b) Log[d]]) Erfc[Sqrt[(-b) Log[d]]/z + Sqrt[(-a) Log[d]] z]), {a, n}] /; Element[n, Integers] && n >= 0

Standard Form

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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> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> </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> <mfrac> <mi> b </mi> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mfrac> </mrow> </msup> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <msqrt> <mi> π </mi> </msqrt> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mfrac> <mrow> <msup> <mo> ∂ </mo> <mi> n </mi> </msup> <mfrac> <mrow> <mrow> <msup> <mi> log </mi> <mrow> <mo> - </mo> <mi> n </mi> </mrow> </msup> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <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> </msqrt> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> erfc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <msqrt> <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> </msqrt> <mi> z </mi> </mfrac> <mo> - </mo> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <msqrt> <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> </msqrt> </mrow> </msup> <mo> ⁢ </mo> <mrow> <mi> erfc </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> + </mo> <mfrac> <msqrt> <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> </msqrt> <mi> z </mi> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <msqrt> <mrow> <mrow> <mo> - </mo> <mi> a </mi> </mrow> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> d </mi> <mo> ) </mo> </mrow> </mrow> </msqrt> </mfrac> </mrow> <mrow> <mo> ∂ </mo> <msup> <mi> a </mi> <mi> n </mi> </msup> </mrow> </mfrac> </mrow> </mrow> <mo> /; 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Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[SuperscriptBox["z_", RowBox[List["2", " ", "n_"]]], " ", SuperscriptBox["d_", RowBox[List[RowBox[List["a_", " ", SuperscriptBox["z_", "2"]]], "+", FractionBox["b_", SuperscriptBox["z_", "2"]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", SqrtBox["\[Pi]"], " ", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["a", ",", "n"]], "}"]]]]], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["Log", "[", "d", "]"]], RowBox[List["-", "n"]]], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "2"]], " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", "d", "]"]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["Log", "[", "d", "]"]]]]]]]], " ", RowBox[List["Erfc", "[", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["Log", "[", "d", "]"]]]]], "z"], "-", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", "d", "]"]]]]], " ", "z"]]]], "]"]]]], "-", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", "d", "]"]]]]], " ", SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["Log", "[", "d", "]"]]]]]]]], " ", RowBox[List["Erfc", "[", RowBox[List[FractionBox[SqrtBox[RowBox[List[RowBox[List["-", "b"]], " ", RowBox[List["Log", "[", "d", "]"]]]]], "z"], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["-", "a"]], " ", RowBox[List["Log", "[", "d", "]"]]]]], " ", "z"]]]], "]"]]]]]], ")"]]]], SqrtBox[RowBox[List[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