Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











Exp






Mathematica Notation

Traditional Notation









Elementary Functions > Exp[z] > Integration > Indefinite integration > Involving functions of the direct function and a power function > Involving products of powers of the direct function and a power function > Involving product of powers of two direct functions and a power function > Involving zn(eb zr+d z)mu (ec zr+f z)nu





http://functions.wolfram.com/01.03.21.0757.01









  


  










Input Form





Integrate[z^n (E^(b z^2 + d z))^\[Mu] (E^(c z^2 + f z))^\[Nu], z] == (-(1/(2 Sqrt[b \[Mu] + c \[Nu]]))) (E^(-((d \[Mu] + f \[Nu])^2/(4 (b \[Mu] + c \[Nu]))) - z (d \[Mu] + b z \[Mu] + f \[Nu] + c z \[Nu])) (E^(z (d + b z)))^\[Mu] (E^(z (f + c z)))^\[Nu] Sum[2^(-n + q) (b \[Mu] + c \[Nu])^(-(1/2) - n) ((-d) \[Mu] - f \[Nu])^(n - q) (d \[Mu] + f \[Nu] + 2 z (b \[Mu] + c \[Nu]))^(1 + q) (-((d \[Mu] + f \[Nu] + 2 z (b \[Mu] + c \[Nu]))^2/ (b \[Mu] + c \[Nu])))^((1/2) (-1 - q)) Binomial[n, q] Gamma[(1 + q)/2, -((d \[Mu] + f \[Nu] + 2 z (b \[Mu] + c \[Nu]))^2/ (4 (b \[Mu] + c \[Nu])))], {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[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["b", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["d", " ", "z"]]]]], ")"]], "\[Mu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["c", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["f", " ", "z"]]]]], ")"]], "\[Nu]"], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[RowBox[List["-", FractionBox["1", RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]]]]]]]], RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", "\[Mu]"]], "+", RowBox[List["f", " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]]]]], "-", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "\[Mu]"]], "+", RowBox[List["b", " ", "z", " ", "\[Mu]"]], "+", RowBox[List["f", " ", "\[Nu]"]], "+", RowBox[List["c", " ", "z", " ", "\[Nu]"]]]], ")"]]]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["b", " ", "z"]]]], ")"]]]]], ")"]], "\[Mu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]], ")"]], "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "d"]], " ", "\[Mu]"]], "-", RowBox[List["f", " ", "\[Nu]"]]]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", "\[Mu]"]], "+", RowBox[List["f", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", "\[Mu]"]], "+", RowBox[List["f", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]]]], ")"]], "2"], RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", "\[Mu]"]], "+", RowBox[List["f", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]]]]]]], "]"]]]]]]]], ")"]]]]]], "/;", 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> &#8747; </mo> <mrow> <mrow> <msup> <mi> z </mi> <mi> n </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </msup> <mo> ) </mo> </mrow> <mi> &#956; </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mi> c </mi> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> f </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> </msup> <mo> ) </mo> </mrow> <mi> &#957; </mi> </msup> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <mn> 1 </mn> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msqrt> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> </msqrt> </mrow> </mfrac> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> f </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> <mo> - </mo> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> f </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> d </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ) </mo> </mrow> <mi> &#956; </mi> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <msup> <mi> &#8519; </mi> <mrow> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> f </mi> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> z </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> ) </mo> </mrow> <mi> &#957; </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </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> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </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> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mrow> <mo> - </mo> <mi> d </mi> </mrow> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> - </mo> <mrow> <mi> f </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> q </mi> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> f </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> q </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> f </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> </mfrac> </mrow> <mo> ) </mo> </mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> q </mi> </mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </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[&quot;(&quot;, GridBox[List[List[TagBox[&quot;n&quot;, Identity]], List[TagBox[&quot;q&quot;, Identity]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]]] </annotation> </semantics> <mo> &#8290; </mo> <mrow> <mi> &#915; </mi> <mo> &#8289; </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> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> d </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> f </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> z </mi> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> b </mi> <mo> &#8290; </mo> <mi> &#956; </mi> </mrow> <mo> + </mo> <mrow> <mi> c </mi> <mo> &#8290; </mo> <mi> &#957; </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mfrac> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </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 /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> d </ci> <ci> z </ci> </apply> </apply> </apply> <ci> &#956; </ci> </apply> <apply> <power /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <ci> c </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <ci> f </ci> <ci> z </ci> </apply> </apply> </apply> <ci> &#957; </ci> </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> <plus /> <apply> <times /> <ci> b </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> c </ci> <ci> &#957; </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> d </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> f </ci> <ci> &#957; </ci> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> c </ci> <ci> &#957; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> d </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> b </ci> <ci> z </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> f </ci> <ci> &#957; </ci> </apply> <apply> <times /> <ci> c </ci> <ci> z </ci> <ci> &#957; </ci> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> <apply> <plus /> <ci> d </ci> <apply> <times /> <ci> b </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <ci> &#956; </ci> </apply> <apply> <power /> <apply> <power /> <exponentiale /> <apply> <times /> <ci> z </ci> <apply> <plus /> <ci> f </ci> <apply> <times /> <ci> c </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <ci> &#957; </ci> </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> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> c </ci> <ci> &#957; </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> <plus /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> d </ci> </apply> <ci> &#956; </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> f </ci> <ci> &#957; </ci> </apply> </apply> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> q </ci> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> d </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> f </ci> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> c </ci> <ci> &#957; </ci> </apply> </apply> </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 /> <apply> <times /> <ci> d </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> f </ci> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> c </ci> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> c </ci> <ci> &#957; </ci> </apply> </apply> <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> <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 /> <apply> <times /> <ci> d </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> f </ci> <ci> &#957; </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> c </ci> <ci> &#957; </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <plus /> <apply> <times /> <ci> b </ci> <ci> &#956; </ci> </apply> <apply> <times /> <ci> c </ci> <ci> &#957; </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </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[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["b_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["d_", " ", "z_"]]]]], ")"]], "\[Mu]_"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["c_", " ", SuperscriptBox["z_", "2"]]], "+", RowBox[List["f_", " ", "z_"]]]]], ")"]], "\[Nu]_"]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", "\[Mu]"]], "+", RowBox[List["f", " ", "\[Nu]"]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]]]]], "-", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["d", " ", "\[Mu]"]], "+", RowBox[List["b", " ", "z", " ", "\[Mu]"]], "+", RowBox[List["f", " ", "\[Nu]"]], "+", RowBox[List["c", " ", "z", " ", "\[Nu]"]]]], ")"]]]]]]], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["d", "+", RowBox[List["b", " ", "z"]]]], ")"]]]]], ")"]], "\[Mu]"], " ", SuperscriptBox[RowBox[List["(", SuperscriptBox["\[ExponentialE]", RowBox[List["z", " ", RowBox[List["(", RowBox[List["f", "+", RowBox[List["c", " ", "z"]]]], ")"]]]]], ")"]], "\[Nu]"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["q", "=", "0"]], "n"], RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "n"]], "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]], RowBox[List[RowBox[List["-", FractionBox["1", "2"]]], "-", "n"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "d"]], " ", "\[Mu]"]], "-", RowBox[List["f", " ", "\[Nu]"]]]], ")"]], RowBox[List["n", "-", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", "\[Mu]"]], "+", RowBox[List["f", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]]]], ")"]], RowBox[List["1", "+", "q"]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", "\[Mu]"]], "+", RowBox[List["f", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]]]], ")"]], "2"], RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]]]]], ")"]], RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", "q"]], ")"]]]]], " ", RowBox[List["Binomial", "[", RowBox[List["n", ",", "q"]], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox[RowBox[List["1", "+", "q"]], "2"], ",", RowBox[List["-", FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List["d", " ", "\[Mu]"]], "+", RowBox[List["f", " ", "\[Nu]"]], "+", RowBox[List["2", " ", "z", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]]]], ")"]], "2"], RowBox[List["4", " ", RowBox[List["(", RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]], ")"]]]]]]]]], "]"]]]]]]]], RowBox[List["2", " ", SqrtBox[RowBox[List[RowBox[List["b", " ", "\[Mu]"]], "+", RowBox[List["c", " ", "\[Nu]"]]]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2002-12-18





© 1998- Wolfram Research, Inc.