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/07.41.20.0016.01

 Input Form

 D[ParabolicCylinderD[\[Nu], z], {z, n}] == (2^(\[Nu]/2 - n) Sqrt[Pi] n! ((1/(z^n Gamma[(1 - \[Nu])/2])) Sum[(((-1)^k k! z^(2 k))/((n - k)! (2 k - n)!)) Sum[(((-1)^j 2^j Pochhammer[-(\[Nu]/2), j])/(j! (k - j)! Pochhammer[1/2, j])) Hypergeometric1F1[-(\[Nu]/2) + j, 1/2 + j, z^2/2], {j, 0, k}], {k, 0, n}] - (Sqrt[2]/Gamma[-(\[Nu]/2)]) (z^(1 + n) Sum[(((-1)^(-j + n) 2^j Pochhammer[(1 - \[Nu])/2, j])/ (j! (n - j)! Pochhammer[3/2, j])) Hypergeometric1F1[ (1 - \[Nu])/2 + j, 3/2 + j, z^2/2], {j, 0, n}] + (n + 1) z^(1 - n) Sum[(z^(2 k)/((2 k - n + 1)! (n - k)!)) Sum[(((-1)^(-j + k) 2^j Binomial[k, j] Pochhammer[(1 - \[Nu])/2, j])/Pochhammer[3/2, j]) Hypergeometric1F1[(1 - \[Nu])/2 + j, 3/2 + j, z^2/2], {j, 0, k}], {k, 0, n - 1}])))/E^(z^2/4) /; Element[n, Integers] && n >= 0

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["{", RowBox[List["z", ",", "n"]], "}"]]], RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["\[Nu]", "2"], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["n", "!"]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List["-", "n"]]], RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["k", "!"]], SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "n"]], ")"]], "!"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["2", "j"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "j"]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]]]]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], "+", "j"]], ",", RowBox[List[FractionBox["1", "2"], "+", "j"]], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]]]]]]]]]], "-", RowBox[List[FractionBox[SqrtBox["2"], RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]]], RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["1", "+", "n"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], SuperscriptBox["2", "j"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "j"]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j"]], ")"]], "!"]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "j"]], "]"]]]]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "+", "j"]], ",", RowBox[List[FractionBox["3", "2"], "+", "j"]], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]]]]]], "+", " ", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], SuperscriptBox["z", RowBox[List["1", "-", "n"]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], RowBox[List[FractionBox[SuperscriptBox["z", RowBox[List["2", " ", "k"]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "-", "n", "+", "1"]], ")"]], "!"]], RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]]]]], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "k"]]], " ", SuperscriptBox["2", "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "j"]], "]"]]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "j"]], "]"]]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "+", "j"]], ",", RowBox[List[FractionBox["3", "2"], "+", "j"]], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]]]]]]]]]]]], ")"]]]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]

 MathML Form

 n D TagBox["D", ParabolicCylinderD] ν ( z ) z n 2 ν 2 - n - z 2 4 π n ! ( z - n Γ ( 1 - ν 2 ) k = 0 n ( - 1 ) k k ! z 2 k ( n - k ) ! ( 2 k - n ) ! j = 0 k ( - 1 ) j 2 j ( - ν 2 ) j TagBox[SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ")"]], "j"], Pochhammer] j ! ( k - j ) ! ( 1 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "j"], Pochhammer] 1 F 1 ( j - ν 2 ; j + 1 2 ; z 2 2 ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["j", "-", FractionBox["\[Nu]", "2"]]], Hypergeometric1F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["j", "+", FractionBox["1", "2"]]], Hypergeometric1F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "2"], Hypergeometric1F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]] - 2 Γ ( - ν 2 ) ( ( n + 1 ) z 1 - n k = 0 n - 1 z 2 k ( 2 k - n + 1 ) ! ( n - k ) ! j = 0 k ( - 1 ) k - j 2 j ( k j ) TagBox[RowBox[List["(", GridBox[List[List[TagBox["k", Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox["j", Identity, Rule[Editable, True], Rule[Selectable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] ( 1 - ν 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ")"]], "j"], Pochhammer] ( 3 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "j"], Pochhammer] 1 F 1 ( j + 1 - ν 2 ; j + 3 2 ; z 2 2 ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["j", "+", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], Hypergeometric1F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["j", "+", FractionBox["3", "2"]]], Hypergeometric1F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "2"], Hypergeometric1F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]] + z n + 1 j = 0 n ( - 1 ) n - j 2 j ( 1 - ν 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ")"]], "j"], Pochhammer] j ! ( n - j ) ! ( 3 2 ) j TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "j"], Pochhammer] 1 F 1 ( j + 1 - ν 2 ; j + 3 2 ; z 2 2 ) TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", "1"], SubscriptBox["F", "1"]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["j", "+", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"]]], Hypergeometric1F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[TagBox[TagBox[RowBox[List["j", "+", FractionBox["3", "2"]]], Hypergeometric1F1, Rule[Editable, True], Rule[Selectable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False], Rule[Selectable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "2"], Hypergeometric1F1, Rule[Editable, True], Rule[Selectable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False], Rule[Selectable, False]] ) ) /; n Condition z n ParabolicCylinderD ν z 2 ν 2 -1 -1 n -1 z 2 4 -1 1 2 n z -1 n Gamma 1 -1 ν 2 -1 -1 k 0 n -1 k k z 2 k n -1 k 2 k -1 n -1 j 0 k -1 j 2 j Pochhammer -1 ν 2 -1 j j k -1 j Pochhammer 1 2 j -1 HypergeometricPFQ j -1 ν 2 -1 j 1 2 z 2 2 -1 -1 2 1 2 Gamma -1 ν 2 -1 -1 n 1 z 1 -1 n k 0 n -1 z 2 k 2 k -1 n 1 n -1 k -1 j 0 k -1 k -1 j 2 j Binomial k j Pochhammer 1 -1 ν 2 -1 j Pochhammer 3 2 j -1 HypergeometricPFQ j 1 -1 ν 2 -1 j 3 2 z 2 2 -1 z n 1 j 0 n -1 n -1 j 2 j Pochhammer 1 -1 ν 2 -1 j j n -1 j Pochhammer 3 2 j -1 HypergeometricPFQ j 1 -1 ν 2 -1 j 3 2 z 2 2 -1 n [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List[RowBox[List["{", RowBox[List["z_", ",", "n_"]], "}"]]]]], RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[SuperscriptBox["2", RowBox[List[FractionBox["\[Nu]", "2"], "-", "n"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]], " ", SqrtBox["\[Pi]"], " ", RowBox[List["n", "!"]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List["k", "!"]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]]]], ")"]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox["2", "j"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "j"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], "+", "j"]], ",", RowBox[List[FractionBox["1", "2"], "+", "j"]], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "j"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "j"]], "]"]]]]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "n"]], ")"]], "!"]]]]]]]]], RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]]], "-", FractionBox[RowBox[List[SqrtBox["2"], " ", RowBox[List["(", RowBox[List[RowBox[List[SuperscriptBox["z", RowBox[List["1", "+", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "n"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "n"]]], " ", SuperscriptBox["2", "j"], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "j"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "+", "j"]], ",", RowBox[List[FractionBox["3", "2"], "+", "j"]], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]], RowBox[List[RowBox[List["j", "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "j"]], ")"]], "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "j"]], "]"]]]]]]]]], "+", RowBox[List[RowBox[List["(", RowBox[List["n", "+", "1"]], ")"]], " ", SuperscriptBox["z", RowBox[List["1", "-", "n"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], RowBox[List["n", "-", "1"]]], FractionBox[RowBox[List[SuperscriptBox["z", RowBox[List["2", " ", "k"]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List[RowBox[List["-", "j"]], "+", "k"]]], " ", SuperscriptBox["2", "j"], " ", RowBox[List["Binomial", "[", RowBox[List["k", ",", "j"]], "]"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "j"]], "]"]]]], ")"]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "+", "j"]], ",", RowBox[List[FractionBox["3", "2"], "+", "j"]], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]], RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "j"]], "]"]]]]]]], RowBox[List[RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "-", "n", "+", "1"]], ")"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["n", "-", "k"]], ")"]], "!"]]]]]]]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]]]]], ")"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]

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

 2007-05-02