Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site Email Comments

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
 











variants of this functions
HermiteH




Mathematica Notation

Traditional Notation









Hypergeometric Functions > HermiteH[nu,z] > Integration > Definite integration > Involving the direct function





http://functions.wolfram.com/07.01.21.0018.01









  


  








Input Form





Integrate[(HermiteH[l, t] HermiteH[m, t] HermiteH[n, t])/E^t^2, {t, -Infinity, Infinity}] == (2^((1/2) (l + m + n)) l! m! n! Sqrt[Pi])/ (((1/2) (l + m - n))! ((1/2) (m + n - l))! ((1/2) (n + l - m))!) /; Element[l, Integers] && l >= 0 && Element[m, Integers] && m >= 0 && Element[n, Integers] && n >= 0 && Element[(1/2) (l + m + n), Integers] && l + m >= n && m + n >= l && n + l >= m








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["t", "2"]]]], " ", RowBox[List["HermiteH", "[", RowBox[List["l", ",", "t"]], "]"]], " ", RowBox[List["HermiteH", "[", RowBox[List["m", ",", "t"]], "]"]], " ", RowBox[List["HermiteH", "[", RowBox[List["n", ",", "t"]], "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["l", "+", "m", "+", "n"]], ")"]]]]], " ", RowBox[List["l", "!"]], " ", RowBox[List["m", "!"]], " ", RowBox[List["n", "!"]], " ", SqrtBox["\[Pi]"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["l", "+", "m", "-", "n"]], ")"]]]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["m", "+", "n", "-", "l"]], ")"]]]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "+", "l", "-", "m"]], ")"]]]], ")"]], "!"]]]]]]], "/;", RowBox[List[RowBox[List["l", "\[Element]", "Integers"]], "&&", RowBox[List["l", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["l", "+", "m", "+", "n"]], ")"]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["l", "+", "m"]], "\[GreaterEqual]", "n"]], "&&", RowBox[List[RowBox[List["m", "+", "n"]], "\[GreaterEqual]", "l"]], "&&", RowBox[List[RowBox[List["n", "+", "l"]], "\[GreaterEqual]", "m"]]]]]]]]








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> <msubsup> <mo> &#8747; </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mi> &#8734; </mi> </msubsup> <mrow> <mrow> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mi> l </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mi> m </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mfrac> <mrow> <msup> <mn> 2 </mn> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> l </mi> <mo> + </mo> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </msup> <mo> &#8290; </mo> <mrow> <mi> l </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> m </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> </mrow> <mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> l </mi> <mo> + </mo> <mi> m </mi> <mo> - </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> l </mi> </mrow> <mo> + </mo> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> l </mi> <mo> - </mo> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> l </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> m </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <semantics> <mi> &#8469; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalN]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> l </mi> <mo> + </mo> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> l </mi> <mo> + </mo> <mi> m </mi> </mrow> <mo> &#8805; </mo> <mi> n </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> m </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> &#8805; </mo> <mi> l </mi> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mi> l </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> &#8805; </mo> <mi> m </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <ci> HermiteH </ci> <ci> l </ci> <ci> t </ci> </apply> <apply> <ci> HermiteH </ci> <ci> m </ci> <ci> t </ci> </apply> <apply> <ci> HermiteH </ci> <ci> n </ci> <ci> t </ci> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> l </ci> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <factorial /> <ci> l </ci> </apply> <apply> <factorial /> <ci> m </ci> </apply> <apply> <factorial /> <ci> n </ci> </apply> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> l </ci> <ci> m </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> </apply> </apply> </apply> <apply> <factorial /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> l </ci> </apply> <ci> m </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <factorial /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> l </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <ci> n </ci> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> l </ci> <integers /> </apply> <apply> <in /> <ci> m </ci> <integers /> </apply> <apply> <in /> <ci> n </ci> <integers /> </apply> <apply> <in /> <apply> <times /> <cn type='rational'> 1 <sep /> 2 </cn> <apply> <plus /> <ci> l </ci> <ci> m </ci> <ci> n </ci> </apply> </apply> <integers /> </apply> <apply> <geq /> <apply> <plus /> <ci> l </ci> <ci> m </ci> </apply> <ci> n </ci> </apply> <apply> <geq /> <apply> <plus /> <ci> m </ci> <ci> n </ci> </apply> <ci> l </ci> </apply> <apply> <geq /> <apply> <plus /> <ci> l </ci> <ci> n </ci> </apply> <ci> m </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["t_", "2"]]]], " ", RowBox[List["HermiteH", "[", RowBox[List["l_", ",", "t_"]], "]"]], " ", RowBox[List["HermiteH", "[", RowBox[List["m_", ",", "t_"]], "]"]], " ", RowBox[List["HermiteH", "[", RowBox[List["n_", ",", "t_"]], "]"]]]], RowBox[List["\[DifferentialD]", "t_"]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["l", "+", "m", "+", "n"]], ")"]]]]], " ", RowBox[List["l", "!"]], " ", RowBox[List["m", "!"]], " ", RowBox[List["n", "!"]], " ", SqrtBox["\[Pi]"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["l", "+", "m", "-", "n"]], ")"]]]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["m", "+", "n", "-", "l"]], ")"]]]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["n", "+", "l", "-", "m"]], ")"]]]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["l", "\[Element]", "Integers"]], "&&", RowBox[List["l", "\[GreaterEqual]", "0"]], "&&", RowBox[List["m", "\[Element]", "Integers"]], "&&", RowBox[List["m", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]], "&&", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List["l", "+", "m", "+", "n"]], ")"]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["l", "+", "m"]], "\[GreaterEqual]", "n"]], "&&", RowBox[List[RowBox[List["m", "+", "n"]], "\[GreaterEqual]", "l"]], "&&", RowBox[List[RowBox[List["n", "+", "l"]], "\[GreaterEqual]", "m"]]]]]]]]]]


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





2001-10-29