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; }

 ParabolicCylinderD

 http://functions.wolfram.com/07.41.06.0013.01

 Input Form

 ParabolicCylinderD[\[Nu], z] \[Proportional] Piecewise[{{(z^\[Nu] (1 - ((-1 + \[Nu]) \[Nu])/(2 z^2) + ((-3 + \[Nu]) (-2 + \[Nu]) (-1 + \[Nu]) \[Nu])/(8 z^4) + O[z^(-6)]))/ E^(z^2/4), Inequality[-(Pi/2), Less, Arg[z], LessEqual, Pi/2]}, {(z^\[Nu] (1 - ((-1 + \[Nu]) \[Nu])/(2 z^2) + ((-3 + \[Nu]) (-2 + \[Nu]) (-1 + \[Nu]) \[Nu])/(8 z^4) + O[z^(-6)]))/ E^(z^2/4) - ((E^(z^2/4 - I Pi \[Nu]) Sqrt[2 Pi] z^(-1 - \[Nu]))/ Gamma[-\[Nu]]) (1 + ((1 + \[Nu]) (2 + \[Nu]))/(2 z^2) + ((1 + \[Nu]) (2 + \[Nu]) (3 + \[Nu]) (4 + \[Nu]))/(8 z^4) + O[z^(-6)]), Arg[z] <= -(Pi/2)}}, (z^\[Nu] (1 - ((-1 + \[Nu]) \[Nu])/(2 z^2) + ((-3 + \[Nu]) (-2 + \[Nu]) (-1 + \[Nu]) \[Nu])/(8 z^4) + O[z^(-6)]))/ E^(z^2/4) - ((E^(z^2/4 + I Pi \[Nu]) Sqrt[2 Pi] z^(-1 - \[Nu]))/ Gamma[-\[Nu]]) (1 + ((1 + \[Nu]) (2 + \[Nu]))/(2 z^2) + ((1 + \[Nu]) (2 + \[Nu]) (3 + \[Nu]) (4 + \[Nu]))/(8 z^4) + O[z^(-6)])] /; (Abs[z] -> Infinity)

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]", ",", " ", "z"]], "]"]], " ", "\[Proportional]", " ", RowBox[List["Piecewise", "[", "\n", " ", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List[SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]], RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], RowBox[List["2", " ", 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["8", " ", 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["z", "\[Nu]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]], RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], RowBox[List["2", " ", 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["8", " ", SuperscriptBox["z", "4"]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", RowBox[List["-", "6"]]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[SuperscriptBox["z", "2"], "4"], "-", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[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["2", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List["(", 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["8", " ", SuperscriptBox["z", "4"]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", RowBox[List["-", "6"]]], "]"]]]], ")"]]]]]], ",", " ", "\n", " ", RowBox[List[RowBox[List["Arg", "[", "z", "]"]], "\[LessEqual]", RowBox[List["-", FractionBox["\[Pi]", "2"]]]]]]], "}"]]]], "}"]], ",", " ", "\n", " ", RowBox[List[RowBox[List[SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]], RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], RowBox[List["2", " ", 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["8", " ", SuperscriptBox["z", "4"]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", RowBox[List["-", "6"]]], "]"]]]], ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox[SuperscriptBox["z", "2"], "4"], "+", RowBox[List["\[ImaginaryI]", " ", "\[Pi]", " ", "\[Nu]"]]]]], " ", SqrtBox[RowBox[List["2", " ", "\[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["2", " ", SuperscriptBox["z", "2"]]]], "+", FractionBox[RowBox[List["(", 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["8", " ", SuperscriptBox["z", "4"]]]], "+", RowBox[List["O", "[", SuperscriptBox["z", RowBox[List["-", "6"]]], "]"]]]], ")"]]]]]]]], "]"]]]], " ", "/;", " ", RowBox[List["(", RowBox[List[RowBox[List["Abs", "[", "z", "]"]], " ", "->", " ", "\[Infinity]"]], ")"]]]]]]

 MathML Form

 D TagBox["D", ParabolicCylinderD] ν ( z ) z ν - z 2 4 ( 1 - ( ν - 1 ) ν 2 z 2 + ( ν - 3 ) ( ν - 2 ) ( ν - 1 ) ν 8 z 4 + O ( 1 z 6 ) ) - π 2 < arg ( z ) π 2 z ν - z 2 4 ( 1 - ( ν - 1 ) ν 2 z 2 + ( ν - 3 ) ( ν - 2 ) ( ν - 1 ) ν 8 z 4 + O ( 1 z 6 ) ) - z 2 4 - π ν 2 π z - ν - 1 Γ ( - ν ) ( 1 + ( ν + 1 ) ( ν + 2 ) 2 z 2 + ( ν + 1 ) ( ν + 3 ) ( ν + 4 ) ( ν + 2 ) 8 z 4 + O ( 1 z 6 ) ) arg ( z ) - π 2 z ν - z 2 4 ( 1 - ( ν - 1 ) ν 2 z 2 + ( ν - 3 ) ( ν - 2 ) ( ν - 1 ) ν 8 z 4 + O ( 1 z 6 ) ) - z 2 4 + π ν 2 π z - ν - 1 Γ ( - ν ) ( 1 + ( ν + 1 ) ( ν + 2 ) 2 z 2 + ( ν + 1 ) ( ν + 3 ) ( ν + 4 ) ( ν + 2 ) 8 z 4 + O ( 1 z 6 ) ) True TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] /; ( "\[LeftBracketingBar]" z "\[RightBracketingBar]" "\[Rule]" ) Condition Proportional ParabolicCylinderD ν z z ν -1 z 2 4 -1 1 -1 ν -1 ν 2 z 2 -1 ν -3 ν -2 ν -1 ν 8 z 4 -1 O 1 z 6 -1 Inequality -1 2 -1 z 2 -1 z ν -1 z 2 4 -1 1 -1 ν -1 ν 2 z 2 -1 ν -3 ν -2 ν -1 ν 8 z 4 -1 O 1 z 6 -1 -1 z 2 4 -1 -1 ν 2 1 2 z -1 ν -1 Gamma -1 ν -1 1 ν 1 ν 2 2 z 2 -1 ν 1 ν 3 ν 4 ν 2 8 z 4 -1 O 1 z 6 -1 z -1 2 -1 z ν -1 z 2 4 -1 1 -1 ν -1 ν 2 z 2 -1 ν -3 ν -2 ν -1 ν 8 z 4 -1 O 1 z 6 -1 -1 z 2 4 -1 ν 2 1 2 z -1 ν -1 Gamma -1 ν -1 1 ν 1 ν 2 2 z 2 -1 ν 1 ν 3 ν 4 ν 2 8 z 4 -1 O 1 z 6 -1 Rule z [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List[RowBox[List[SuperscriptBox["z", "\[Nu]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]], " ", RowBox[List["(", RowBox[List["1", "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["-", "1"]], "+", "\[Nu]"]], ")"]], " ", "\[Nu]"]], RowBox[List["2", " ", 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["8", " ", SuperscriptBox["z", "4"]]]], "+", FractionBox["1", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], "6"]]]], ")"]]]], 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 Date Added to functions.wolfram.com (modification date)

 2007-05-02