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http://functions.wolfram.com/01.03.21.0223.01
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Integrate[z^(2 n) d^(a z^2 + b/z^2 + c), z] ==
((d^c 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
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<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> <mo> + </mo> <mi> c </mi> </mrow> </msup> </mrow> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> d </mi> <mi> c </mi> </msup> <mo> ⁢ </mo> <msqrt> <mi> π </mi> </msqrt> </mrow> <mo> ) </mo> </mrow> <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> /; </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> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </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> <apply> <power /> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <ci> c </ci> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <times /> <apply> <power /> <ci> d </ci> <ci> c </ci> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <partialdiff /> <bvar> <ci> a </ci> <degree> <ci> n </ci> </degree> </bvar> <apply> <times /> <apply> <power /> <apply> <ln /> <ci> d </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -2 </cn> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <ln /> <ci> d </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <ln /> <ci> d </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> Erfc </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <ln /> <ci> d </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <ln /> <ci> d </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <ln /> <ci> d </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <ln /> <ci> d </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> <apply> <ci> Erfc </ci> <apply> <plus /> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <ln /> <ci> d </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <ci> z </ci> </apply> <apply> <times /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> b </ci> </apply> <apply> <ln /> <ci> d </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> a </ci> </apply> <apply> <ln /> <ci> d </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>
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| 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"]], "+", "c_"]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["d", "c"], " ", 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"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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