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 HermiteH

 http://functions.wolfram.com/05.01.06.0017.01

 Input Form

 HermiteH[n, z] \[Proportional] ((((-1)^Floor[n/2] 2^n)/Sqrt[Pi]) E^(z^2/2) Floor[n/2]! (Cos[Pi ((1 - (-1)^n)/4) - Sqrt[2 n + 1] z] - (z^3/(4 Sqrt[2] Floor[n/2]^(1/2))) (Cos[((-1)^n Pi)/4 + Sqrt[2 n + 1] z] - Sin[((-1)^n Pi)/4 + Sqrt[2 n + 1] z]) + ((32 z^2 (-1 - 2 z^2 + (-1)^n (-1 + z^2)))/(512 z^2 Floor[n/2])) Cos[Pi ((1 - (-1)^n)/4) - Sqrt[2 n + 1] z] + ((z (-5 - 4 z^2 + 2 (-1)^n (2 + z^2)))/(32 Sqrt[2] Floor[n/2]^(3/2))) (Cos[((-1)^n Pi)/4 + Sqrt[2 n + 1] z] - Sin[((-1)^n Pi)/4 + Sqrt[2 n + 1] z]) + \[Ellipsis]))/ Floor[n/2]^((1 + (-1)^n)/4) /; (n -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["HermiteH", "[", RowBox[List["n", ",", "z"]], "]"]], "\[Proportional]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], " ", SuperscriptBox["2", "n"]]], SqrtBox["\[Pi]"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], "2"]], " ", RowBox[List[RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "!"]], " ", SuperscriptBox[RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], RowBox[List["-", FractionBox[RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], "4"]]]], RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List["\[Pi]", FractionBox[RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], "4"]]], "-", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], " ", "z"]]]], "]"]], "-", RowBox[List[FractionBox[SuperscriptBox["z", "3"], RowBox[List["4", " ", SqrtBox["2"], " ", SuperscriptBox[RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], RowBox[List["1", "/", "2"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "\[Pi]"]], "4"], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], " ", "z"]]]], "]"]], "-", RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "\[Pi]"]], "4"], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], " ", "z"]]]], "]"]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List["32", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]]]]]], ")"]]]], RowBox[List["512", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List["\[Pi]", FractionBox[RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], "4"]]], "-", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], " ", "z"]]]], "]"]]]], "+", RowBox[List[FractionBox[RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "-", RowBox[List["4", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List["(", RowBox[List["2", "+", SuperscriptBox["z", "2"]]], ")"]]]]]], ")"]]]], RowBox[List["32", " ", SqrtBox["2"], " ", SuperscriptBox[RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], RowBox[List["3", "/", "2"]]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "\[Pi]"]], "4"], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], " ", "z"]]]], "]"]], "-", RowBox[List["Sin", "[", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "\[Pi]"]], "4"], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], " ", "z"]]]], "]"]]]], ")"]]]], "+", "\[Ellipsis]"]], ")"]]]]]], "/;", RowBox[List["(", RowBox[List["n", "\[Rule]", "\[Infinity]"]], ")"]]]]]]

 MathML Form

 H n ( z ) ( - 1 ) n 2 2 n π z 2 2 n 2 ! n 2 - 1 + ( - 1 ) n 4 ( cos ( π ( 1 - ( - 1 ) n ) 4 - 2 n + 1 z ) - z 3 4 2 n 2 ( cos ( 2 n + 1 z + ( - 1 ) n π 4 ) - sin ( 2 n + 1 z + ( - 1 ) n π 4 ) ) + 32 z 2 ( - 2 z 2 + ( - 1 ) n ( z 2 - 1 ) - 1 ) 512 z 2 n 2 cos ( π ( 1 - ( - 1 ) n ) 4 - 2 n + 1 z ) + z ( - 4 z 2 + 2 ( - 1 ) n ( z 2 + 2 ) - 5 ) 32 2 n 2 3 / 2 ( cos ( 2 n + 1 z + ( - 1 ) n π 4 ) - sin ( 2 n + 1 z + ( - 1 ) n π 4 ) ) + ) /; ( n "\[Rule]" ) Condition Proportional HermiteH n z -1 n 2 -1 2 n 1 2 -1 z 2 2 -1 n 2 -1 n 2 -1 -1 1 -1 n 4 -1 1 -1 -1 n 4 -1 -1 2 n 1 1 2 z -1 z 3 4 2 1 2 n 2 -1 1 2 -1 2 n 1 1 2 z -1 n 4 -1 -1 sin 2 n 1 1 2 z -1 n 4 -1 32 z 2 -2 z 2 -1 n z 2 -1 -1 512 z 2 n 2 -1 -1 1 -1 -1 n 4 -1 -1 2 n 1 1 2 z z -4 z 2 2 -1 n z 2 2 -5 32 2 1 2 n 2 -1 3 2 -1 2 n 1 1 2 z -1 n 4 -1 -1 2 n 1 1 2 z -1 n 4 -1 Rule n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["HermiteH", "[", RowBox[List["n_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]], " ", SuperscriptBox["2", "n"]]], ")"]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["z", "2"], "2"]], " ", RowBox[List[RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], "!"]], " ", SuperscriptBox[RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], RowBox[List[RowBox[List["-", FractionBox["1", "4"]]], " ", RowBox[List["(", RowBox[List["1", "+", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]]]]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]]]], "-", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], " ", "z"]]]], "]"]], "-", FractionBox[RowBox[List[SuperscriptBox["z", "3"], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "\[Pi]"]], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], " ", "z"]]]], "]"]], "-", RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "\[Pi]"]], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], " ", "z"]]]], "]"]]]], ")"]]]], RowBox[List["4", " ", SqrtBox["2"], " ", SqrtBox[RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["32", " ", SuperscriptBox["z", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "-", RowBox[List["2", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", SuperscriptBox["z", "2"]]], ")"]]]]]], ")"]]]], ")"]], " ", RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", "\[Pi]", " ", RowBox[List["(", RowBox[List["1", "-", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"]]], ")"]]]], "-", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], " ", "z"]]]], "]"]]]], RowBox[List["512", " ", SuperscriptBox["z", "2"], " ", RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]]]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List["z", " ", RowBox[List["(", RowBox[List[RowBox[List["-", "5"]], "-", RowBox[List["4", " ", SuperscriptBox["z", "2"]]], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List["2", "+", SuperscriptBox["z", "2"]]], ")"]]]]]], ")"]]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List["Cos", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "\[Pi]"]], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], " ", "z"]]]], "]"]], "-", RowBox[List["Sin", "[", RowBox[List[RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", "\[Pi]"]], "+", RowBox[List[SqrtBox[RowBox[List[RowBox[List["2", " ", "n"]], "+", "1"]]], " ", "z"]]]], "]"]]]], ")"]]]], RowBox[List["32", " ", SqrtBox["2"], " ", SuperscriptBox[RowBox[List["Floor", "[", FractionBox["n", "2"], "]"]], RowBox[List["3", "/", "2"]]]]]], "+", "\[Ellipsis]"]], ")"]]]], SqrtBox["\[Pi]"]], "/;", RowBox[List["(", RowBox[List["n", "\[Rule]", "\[Infinity]"]], ")"]]]]]]]]

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

 2007-05-02