|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/05.01.11.0002.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
HermiteH[n, z] == n! SeriesTerm[E^((4 z^2 t)/(1 + 4 t))/Sqrt[1 + 4 t],
{t, 0, n}] /; Element[2 n, Integers]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HermiteH", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Equal]", RowBox[List[RowBox[List["n", "!"]], RowBox[List["SeriesTerm", "[", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["4", SuperscriptBox["z", "2"], "t"]], RowBox[List["1", "+", RowBox[List["4", "t"]]]]]], SqrtBox[RowBox[List["1", "+", RowBox[List["4", "t"]]]]]], ",", RowBox[List["{", RowBox[List["t", ",", "0", ",", "n"]], "}"]]]], "]"]]]]]], "/;", RowBox[List["Element", "[", RowBox[List[RowBox[List["2", "n"]], ",", " ", "Integers"]], "]"]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msub> <mi> H </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> [ </mo> <msup> <mi> t </mi> <mi> n </mi> </msup> <mtext> </mtext> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mfrac> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> t </mi> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> t </mi> </mrow> </mrow> </mfrac> </msup> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> t </mi> </mrow> </mrow> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <mrow> <mrow> <mrow> <msub> <mi> H </mi> <mi> n </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> [ </mo> <msup> <mi> t </mi> <mi> n </mi> </msup> <mtext> </mtext> <mo> ] </mo> </mrow> <mo> ⁢ </mo> <mfrac> <msup> <mi> ⅇ </mi> <mfrac> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mo> ⁢ </mo> <mi> t </mi> </mrow> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> t </mi> </mrow> </mrow> </mfrac> </msup> <msqrt> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> t </mi> </mrow> </mrow> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> </mrow> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HermiteH", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["n", "!"]], " ", RowBox[List["SeriesTerm", "[", RowBox[List[FractionBox[SuperscriptBox["\[ExponentialE]", FractionBox[RowBox[List["4", " ", SuperscriptBox["z", "2"], " ", "t"]], RowBox[List["1", "+", RowBox[List["4", " ", "t"]]]]]], SqrtBox[RowBox[List["1", "+", RowBox[List["4", " ", "t"]]]]]], ",", RowBox[List["{", RowBox[List["t", ",", "0", ",", "n"]], "}"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["2", " ", "n"]], "\[Element]", "Integers"]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
){kind=small}
