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http://functions.wolfram.com/05.01.07.0004.01
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HermiteH[n, z] == (2^n/Sqrt[Pi]) Integrate[(z + I t)^n/E^t^2,
{t, -Infinity, Infinity}]
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Cell[BoxData[RowBox[List[RowBox[List["HermiteH", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[FractionBox[RowBox[List[" ", SuperscriptBox["2", "n"], " "]], SqrtBox["\[Pi]"]], RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], "\[Infinity]"], " ", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["t", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", RowBox[List["\[ImaginaryI]", " ", "t"]]]], ")"]], "n"], RowBox[List["\[DifferentialD]", "t"]]]]]]]]]]]]
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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> <msub> <mi> H </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mrow> <mtext> </mtext> <msup> <mn> 2 </mn> <mi> n </mi> </msup> </mrow> <msqrt> <mi> π </mi> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <msubsup> <mo> ∫ </mo> <mrow> <mo> - </mo> <mi> ∞ </mi> </mrow> <mi> ∞ </mi> </msubsup> <mrow> <msup> <mi> ⅇ </mi> <mrow> <mo> - </mo> <msup> <mi> t </mi> <mn> 2 </mn> </msup> </mrow> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> + </mo> <mrow> <mi> ⅈ </mi> <mo> ⁢ </mo> <mi> t </mi> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> t </mi> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> HermiteH </ci> <ci> n </ci> <ci> z </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <apply> <power /> <apply> <power /> <pi /> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </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> <power /> <exponentiale /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <power /> <ci> t </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <imaginaryi /> <ci> t </ci> </apply> </apply> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HermiteH", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[SuperscriptBox["2", "n"], " ", RowBox[List[SubsuperscriptBox["\[Integral]", RowBox[List["-", "\[Infinity]"]], "\[Infinity]"], RowBox[List[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List["-", SuperscriptBox["t", "2"]]]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "+", RowBox[List["\[ImaginaryI]", " ", "t"]]]], ")"]], "n"]]], RowBox[List["\[DifferentialD]", "t"]]]]]]]], SqrtBox["\[Pi]"]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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