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









Polynomials > HermiteH[n,z] > Operations > Orthogonality, completeness, and Fourier expansions





http://functions.wolfram.com/05.01.25.0003.01









  


  










Input Form





Sum[((1/Sqrt[Sqrt[Pi] 2^n n!]) Exp[-(x^2/2)] HermiteH[n, x]) ((1/Sqrt[Sqrt[Pi] 2^n n!]) Exp[-(y^2/2)] HermiteH[n, y]), {n, 0, Infinity}] == DiracDelta[x - y] /; Element[x, Reals] && Element[y, Reals]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[FractionBox["1", SqrtBox[RowBox[List[SqrtBox["\[Pi]"], SuperscriptBox["2", "n"], RowBox[List["n", "!"]]]]]], RowBox[List["Exp", "[", RowBox[List["-", FractionBox[SuperscriptBox["x", "2"], "2"]]], "]"]], RowBox[List["HermiteH", "[", RowBox[List["n", ",", "x"]], "]"]]]], ")"]], RowBox[List["(", RowBox[List[FractionBox["1", SqrtBox[RowBox[List[SqrtBox["\[Pi]"], SuperscriptBox["2", "n"], RowBox[List["n", "!"]]]]]], RowBox[List["Exp", "[", RowBox[List["-", FractionBox[SuperscriptBox["y", "2"], "2"]]], "]"]], RowBox[List["HermiteH", "[", RowBox[List["n", ",", "y"]], "]"]]]], ")"]]]]]], "\[Equal]", RowBox[List["DiracDelta", "[", RowBox[List["x", "-", "y"]], "]"]]]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "\[And]", RowBox[List["y", "\[Element]", "Reals"]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <msup> <mi> x </mi> <mn> 2 </mn> </msup> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mn> 1 </mn> <msqrt> <mrow> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msup> <mn> 2 </mn> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> </mrow> </msqrt> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <msup> <mi> y </mi> <mn> 2 </mn> </msup> <mn> 2 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msub> <mi> H </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> y </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#10869; </mo> <mrow> <semantics> <mi> &#948; </mi> <annotation-xml encoding='MathML-Content'> <ci> DiracDelta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> x </mi> <mo> - </mo> <mi> y </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <factorial /> <ci> n </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> x </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> HermiteH </ci> <ci> n </ci> <ci> x </ci> </apply> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <apply> <times /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <factorial /> <ci> n </ci> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> y </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> HermiteH </ci> <ci> n </ci> <ci> y </ci> </apply> </apply> </apply> </apply> <apply> <ci> DiracDelta </ci> <apply> <plus /> <ci> x </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> y </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n_", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["x_", "2"], "2"]]]], " ", RowBox[List["HermiteH", "[", RowBox[List["n_", ",", "x_"]], "]"]]]], ")"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["y_", "2"], "2"]]]], " ", RowBox[List["HermiteH", "[", RowBox[List["n_", ",", "y_"]], "]"]]]], ")"]]]], RowBox[List[SqrtBox[RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox["2", "n_"], " ", RowBox[List["n_", "!"]]]]], " ", SqrtBox[RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox["2", "n_"], " ", RowBox[List["n_", "!"]]]]]]]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["DiracDelta", "[", RowBox[List["x", "-", "y"]], "]"]], "/;", RowBox[List[RowBox[List["x", "\[Element]", "Reals"]], "&&", RowBox[List["y", "\[Element]", "Reals"]]]]]]]]]]










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





2001-10-29