html, body, form { margin: 0; padding: 0; width: 100%; } #calculate { position: relative; width: 177px; height: 110px; background: transparent url(/images/alphabox/embed_functions_inside.gif) no-repeat scroll 0 0; } #i { position: relative; left: 18px; top: 44px; width: 133px; border: 0 none; outline: 0; font-size: 11px; } #eq { width: 9px; height: 10px; background: transparent; position: absolute; top: 47px; right: 18px; cursor: pointer; }

 HermiteH

 http://functions.wolfram.com/07.01.06.0025.01

 Input Form

 HermiteH[\[Nu], z] \[Proportional] Piecewise[{{2^\[Nu] z^\[Nu] (1 - ((-1 + \[Nu]) \[Nu])/(4 z^2) + ((-3 + \[Nu]) (-2 + \[Nu]) (-1 + \[Nu]) \[Nu])/(32 z^4) + O[z^(-6)]), Inequality[-(Pi/2), Less, Arg[z], LessEqual, Pi/2]}, {2^\[Nu] z^\[Nu] (1 - ((-1 + \[Nu]) \[Nu])/(4 z^2) + ((-3 + \[Nu]) (-2 + \[Nu]) (-1 + \[Nu]) \[Nu])/(32 z^4) + O[z^(-6)]) - ((E^(z^2 + I Pi \[Nu]) Sqrt[Pi] z^(-1 - \[Nu]))/ Gamma[-\[Nu]]) (1 + ((1 + \[Nu]) (2 + \[Nu]))/(4 z^2) + ((1 + \[Nu]) (2 + \[Nu]) (3 + \[Nu]) (4 + \[Nu]))/(32 z^4) + O[z^(-6)]), Arg[z] > Pi/2}}, 2^\[Nu] z^\[Nu] (1 - ((-1 + \[Nu]) \[Nu])/(4 z^2) + ((-3 + \[Nu]) (-2 + \[Nu]) (-1 + \[Nu]) \[Nu])/(32 z^4) + O[z^(-6)]) - ((E^(z^2 - I Pi \[Nu]) Sqrt[Pi] z^(-1 - \[Nu]))/Gamma[-\[Nu]]) (1 + ((1 + \[Nu]) (2 + \[Nu]))/(4 z^2) + ((1 + \[Nu]) (2 + \[Nu]) (3 + \[Nu]) (4 + \[Nu]))/(32 z^4) + O[z^(-6)])] /; (Abs[z] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HermiteH", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List["Piecewise", "[", "\n", " ", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], RowBox[List["32", " ", SuperscriptBox["z", "4"]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", RowBox[List["-", "6"]]], "]"]]]], ")"]]]], ",", " ", RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]]], "}"]], ",", " ", "\n", " ", RowBox[List["{", RowBox[List[RowBox[List[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], RowBox[List["32", " ", SuperscriptBox["z", "4"]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", RowBox[List["-", "6"]]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["z", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]]]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "\[Nu]"]], ")"]]]], RowBox[List["32", " ", SuperscriptBox["z", "4"]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", RowBox[List["-", "6"]]], "]"]]]], ")"]]]]]], ",", " ", "\n", " ", RowBox[List[RowBox[List["Arg", "[", "z", "]"]], ">", FractionBox["\[Pi]", "2"]]]]], "}"]]]], "}"]], ",", " ", "\n", " ", RowBox[List[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], RowBox[List["32", " ", SuperscriptBox["z", "4"]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", RowBox[List["-", "6"]]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["z", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]], RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]]]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "\[Nu]"]], ")"]]]], RowBox[List["32", " ", SuperscriptBox["z", "4"]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", RowBox[List["-", "6"]]], "]"]]]], ")"]]]]]]]], "]"]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]

 MathML Form

 H ν ( z ) 2 ν z ν ( 1 - ( ν - 1 ) ν 4 z 2 + ( ν - 3 ) ( ν - 2 ) ( ν - 1 ) ν 32 z 4 + O ( 1 z 6 ) ) - π 2 < arg ( z ) π 2 2 ν z ν ( 1 - ( ν - 1 ) ν 4 z 2 + ( ν - 3 ) ( ν - 2 ) ( ν - 1 ) ν 32 z 4 + O ( 1 z 6 ) ) - z 2 + π ν π z - ν - 1 Γ ( - ν ) ( 1 + ( ν + 1 ) ( ν + 2 ) 4 z 2 + ( ν + 1 ) ( ν + 3 ) ( ν + 4 ) ( ν + 2 ) 32 z 4 + O ( 1 z 6 ) ) arg ( z ) > π 2 2 ν z ν ( 1 - ( ν - 1 ) ν 4 z 2 + ( ν - 3 ) ( ν - 2 ) ( ν - 1 ) ν 32 z 4 + O ( 1 z 6 ) ) - z 2 - π ν π z - ν - 1 Γ ( - ν ) ( 1 + ( ν + 1 ) ( ν + 2 ) 4 z 2 + ( ν + 1 ) ( ν + 3 ) ( ν + 4 ) ( ν + 2 ) 32 z 4 + O ( 1 z 6 ) ) True TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional HermiteH ν z 2 ν z ν 1 -1 ν -1 ν 4 z 2 -1 ν -3 ν -2 ν -1 ν 32 z 4 -1 O 1 z 6 -1 Inequality -1 2 -1 z 2 -1 2 ν z ν 1 -1 ν -1 ν 4 z 2 -1 ν -3 ν -2 ν -1 ν 32 z 4 -1 O 1 z 6 -1 -1 z 2 ν 1 2 z -1 ν -1 Gamma -1 ν -1 1 ν 1 ν 2 4 z 2 -1 ν 1 ν 3 ν 4 ν 2 32 z 4 -1 O 1 z 6 -1 z 2 -1 2 ν z ν 1 -1 ν -1 ν 4 z 2 -1 ν -3 ν -2 ν -1 ν 32 z 4 -1 O 1 z 6 -1 -1 z 2 -1 ν 1 2 z -1 ν -1 Gamma -1 ν -1 1 ν 1 ν 2 4 z 2 -1 ν 1 ν 3 ν 4 ν 2 32 z 4 -1 O 1 z 6 -1 Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HermiteH", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], RowBox[List["32", " ", SuperscriptBox["z", "4"]]]], "+", FractionBox["1", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], "6"]]]], ")"]]]], RowBox[List[RowBox[List["-", FractionBox["\[Pi]", "2"]]], "<", RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", FractionBox["\[Pi]", "2"]]]], List[RowBox[List[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], RowBox[List["32", " ", SuperscriptBox["z", "4"]]]], "+", FractionBox["1", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], "6"]]]], ")"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["z", "2"], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]]]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "\[Nu]"]], ")"]]]], RowBox[List["32", " ", SuperscriptBox["z", "4"]]]], "+", FractionBox["1", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], "6"]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]], RowBox[List[RowBox[List["Arg", "[", "z", "]"]], ">", FractionBox["\[Pi]", "2"]]]], List[RowBox[List[RowBox[List[SuperscriptBox["2", "\[Nu]"], " ", SuperscriptBox["z", "\[Nu]"], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "3"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "2"]], "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], RowBox[List["32", " ", SuperscriptBox["z", "4"]]]], "+", FractionBox["1", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], "6"]]]], ")"]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[SuperscriptBox["z", "2"], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", SqrtBox["\[Pi]"], " ", SuperscriptBox["z", RowBox[List[RowBox[List["-", "1"]], "-", "\[Nu]"]]]]], ")"]], " ", RowBox[List["(", RowBox[List["1", "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]]]], RowBox[List["4", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["1", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["2", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["3", "+", "\[Nu]"]], ")"]], " ", RowBox[List["(", RowBox[List["4", "+", "\[Nu]"]], ")"]]]], RowBox[List["32", " ", SuperscriptBox["z", "4"]]]], "+", FractionBox["1", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], "6"]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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

 2007-05-02