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variants of this functions

Mathematica Notation

Traditional Notation

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




Input Form

f[x] == Sum[Subscript[c, n] Subscript[\[Psi], n][x], {n, 0, Infinity}] /; Subscript[c, n] == Integrate[Subscript[\[Psi], n][t] f[t], {t, -Infinity, Infinity}] && Subscript[\[Psi], n][x] == (1/Sqrt[Sqrt[Pi] 2^n n!]) Exp[-(x^2/2)] HermiteH[n, x] && Element[x, Reals]

Standard Form

Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["f", "[", "x", "]"]], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["c", "n"], RowBox[List[SubscriptBox["\[Psi]", "n"], "[", "x", "]"]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["c", "n"], "\[Equal]", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], "\[Infinity]"], RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "n"], "[", "t", "]"]], RowBox[List["f", "[", "t", "]"]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], "\[And]", RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "n"], "[", "x", "]"]], "\[Equal]", 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"]], "]"]]]]]], "\[And]", RowBox[List["x", "\[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> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> n </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <msub> <mi> c </mi> <mi> n </mi> </msub> <mo> &#8290; </mo> <mrow> <msub> <mi> &#968; </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> c </mi> <mi> n </mi> </msub> <mo> &#10869; </mo> <mrow> <msubsup> <mo> &#8747; </mo> <mrow> <mo> - </mo> <mi> &#8734; </mi> </mrow> <mi> &#8734; </mi> </msubsup> <mrow> <mrow> <mrow> <msub> <mi> &#968; </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> f </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> t </mi> <mo> ) </mo> </mrow> </mrow> <mo> &#8290; </mo> <mrow> <mo> &#8518; </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <msub> <mi> &#968; </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> x </mi> <mo> ) </mo> </mrow> <mo> &#10869; </mo> <mrow> <mfrac> <mrow> <mn> 1 </mn> <mtext> </mtext> </mrow> <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> </mrow> <mo> &#8743; </mo> <mrow> <mi> x </mi> <mo> &#8712; </mo> <semantics> <mi> &#8477; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalR]&quot;, Function[Reals]] </annotation> </semantics> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> f </ci> <ci> x </ci> </apply> <apply> <sum /> <bvar> <ci> n </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> n </ci> </apply> <apply> <apply> <ci> Subscript </ci> <ci> &#968; </ci> <ci> n </ci> </apply> <ci> x </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> c </ci> <ci> n </ci> </apply> <apply> <int /> <bvar> <ci> t </ci> </bvar> <lowlimit> <apply> <times /> <cn type='integer'> -1 </cn> <infinity /> </apply> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> &#968; </ci> <ci> n </ci> </apply> <ci> t </ci> </apply> <apply> <ci> f </ci> <ci> t </ci> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <apply> <ci> Subscript </ci> <ci> &#968; </ci> <ci> n </ci> </apply> <ci> x </ci> </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> 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> <apply> <in /> <ci> x </ci> <reals /> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["f", "[", "x_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["n", "=", "0"]], "\[Infinity]"], RowBox[List[SubscriptBox["c", "n"], " ", RowBox[List[SubscriptBox["\[Psi]", "n"], "[", "x", "]"]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["c", "n"], "\[Equal]", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], "\[Infinity]"], RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "n"], "[", "t", "]"]], " ", RowBox[List["f", "[", "t", "]"]]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], "&&", RowBox[List[RowBox[List[SubscriptBox["\[Psi]", "n"], "[", "x", "]"]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["x", "2"], "2"]]]], " ", RowBox[List["HermiteH", "[", RowBox[List["n", ",", "x"]], "]"]]]], SqrtBox[RowBox[List[SqrtBox["\[Pi]"], " ", SuperscriptBox["2", "n"], " ", RowBox[List["n", "!"]]]]]]]], "&&", RowBox[List["x", "\[Element]", "Reals"]]]]]]]]]]

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