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 ParabolicCylinderD

 http://functions.wolfram.com/07.41.20.0003.01

 Input Form

 D[ParabolicCylinderD[\[Nu], z], \[Nu]] == (1/2) Log[2] ParabolicCylinderD[\[Nu], z] + ((1/(4 Gamma[-\[Nu]])) (Gamma[-(\[Nu]/2)] PolyGamma[(1 - \[Nu])/2] Hypergeometric1F1[-(\[Nu]/2), 1/2, z^2/2] - Sqrt[2] z Gamma[(1 - \[Nu])/2] PolyGamma[-(\[Nu]/2)] Hypergeometric1F1[(1 - \[Nu])/2, 3/2, z^2/2]) - ((2^(\[Nu] - 1/2) Sqrt[Pi] z)/Gamma[-(\[Nu]/2)]) (PolyGamma[(1 - \[Nu])/2] Hypergeometric1F1[(1 - \[Nu])/2, 3/2, z^2/2] - Sum[((z^(2 k) Pochhammer[(1 - \[Nu])/2, k])/ (2^k (k! Pochhammer[3/2, k]))) PolyGamma[(1 - \[Nu])/2 + k], {k, 0, Infinity}]) + ((2^(\[Nu] - 1) Sqrt[Pi])/Gamma[(1 - \[Nu])/2]) (PolyGamma[-(\[Nu]/2)] Hypergeometric1F1[-(\[Nu]/2), 1/2, z^2/2] - Sum[((z^(2 k) Pochhammer[-(\[Nu]/2), k])/(2^k (k! Pochhammer[1/2, k]))) PolyGamma[k - \[Nu]/2], {k, 0, Infinity}]))/(2^(\[Nu]/2) E^(z^2/4))

 Standard Form

 Cell[BoxData[RowBox[List[RowBox[List[SubscriptBox["\[PartialD]", "\[Nu]"], RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "\[Equal]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Log", "[", "2", "]"]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List[RowBox[List["-", "\[Nu]"]], "/", "2"]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]], RowBox[List["(", RowBox[List[RowBox[List[FractionBox["1", RowBox[List["4", " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]], RowBox[List["PolyGamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]], " ", "-", RowBox[List[SqrtBox["2"], " ", "z", " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]], RowBox[List["Hypergeometric1F1", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", FractionBox["3", "2"], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]]]], " ", ")"]]]], "-", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]]], " ", SqrtBox["\[Pi]"], " ", "z", " "]], RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]]], RowBox[List["(", " ", RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]], RowBox[List["Hypergeometric1F1", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", FractionBox["3", "2"], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], InterpretationBox["\[Infinity]", DirectedInfinity[1]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], SuperscriptBox["z", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]]], " ", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "+", "k"]], "]"]]]]]]]], ")"]]]], "+", RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "-", "1"]]], " ", SqrtBox["\[Pi]"]]], RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]]], RowBox[List["(", " ", RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]], RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], InterpretationBox["\[Infinity]", DirectedInfinity[1]]], RowBox[List[FractionBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], SuperscriptBox["z", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", FractionBox["\[Nu]", "2"]]], "]"]]]]]]]], ")"]]]]]], ")"]]]]]]]]]]

 MathML Form

 D TagBox["D", ParabolicCylinderD] ν ( z ) ν 1 2 log ( 2 ) D TagBox["D", ParabolicCylinderD] ν ( z ) + 2 - ν 2 - z 2 4 ( 1 4 Γ ( - ν ) ( Γ ( - ν 2 ) ψ TagBox["\[Psi]", PolyGamma] ( 1 - ν 2 ) 1 F 1 ( - ν 2 ; 1 2 ; z 2 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["1", "2"], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "2"], Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] - 2 z Γ ( 1 - ν 2 ) ψ TagBox["\[Psi]", PolyGamma] ( - ν 2 ) 1 F 1 ( 1 - ν 2 ; 3 2 ; z 2 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["3", "2"], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "2"], Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] ) + 2 ν - 1 π Γ ( 1 - ν 2 ) ( ψ TagBox["\[Psi]", PolyGamma] ( - ν 2 ) 1 F 1 ( - ν 2 ; 1 2 ; z 2 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["-", FractionBox["\[Nu]", "2"]]], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["1", "2"], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "2"], Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] - k = 0 2 - k ( - ν 2 ) k TagBox[RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], SubscriptBox[RowBox[List["(", RowBox[List["-", FractionBox["\[Nu]", "2"]]], ")"]], "k"]]], Pochhammer] ψ TagBox["\[Psi]", PolyGamma] ( k - ν 2 ) z 2 k k ! ( 1 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["1", "2"], ")"]], "k"], Pochhammer] ) - 2 ν - 1 2 π z Γ ( - ν 2 ) ( ψ TagBox["\[Psi]", PolyGamma] ( 1 - ν 2 ) 1 F 1 ( 1 - ν 2 ; 3 2 ; z 2 2 ) TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox["F", FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[FractionBox["3", "2"], Hypergeometric1F1, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1, Rule[Editable, False]], ";", TagBox[FractionBox[SuperscriptBox["z", "2"], "2"], Hypergeometric1F1, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQ[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1] - k = 0 2 - k ( 1 - ν 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ")"]], "k"], Pochhammer] ψ TagBox["\[Psi]", PolyGamma] ( k + 1 - ν 2 ) z 2 k k ! ( 3 2 ) k TagBox[SubscriptBox[RowBox[List["(", FractionBox["3", "2"], ")"]], "k"], Pochhammer] ) ) ν ParabolicCylinderD ν z 1 2 2 ParabolicCylinderD ν z 2 -1 ν 2 -1 -1 z 2 4 -1 1 4 Gamma -1 ν -1 Gamma -1 ν 2 -1 PolyGamma 1 -1 ν 2 -1 Hypergeometric1F1 -1 ν 2 -1 1 2 z 2 2 -1 -1 2 1 2 z Gamma 1 -1 ν 2 -1 PolyGamma -1 ν 2 -1 Hypergeometric1F1 1 -1 ν 2 -1 3 2 z 2 2 -1 2 ν -1 1 2 Gamma 1 -1 ν 2 -1 -1 PolyGamma -1 ν 2 -1 Hypergeometric1F1 -1 ν 2 -1 1 2 z 2 2 -1 -1 k 0 DirectedInfinity 1 Pochhammer 2 -1 k Subscript -1 ν 2 -1 k PolyGamma k -1 ν 2 -1 z 2 k k Pochhammer 1 2 k -1 -1 2 ν -1 1 2 1 2 z Gamma -1 ν 2 -1 -1 PolyGamma 1 -1 ν 2 -1 Hypergeometric1F1 1 -1 ν 2 -1 3 2 z 2 2 -1 -1 k 0 DirectedInfinity 1 2 -1 k Pochhammer 1 -1 ν 2 -1 k PolyGamma k 1 -1 ν 2 -1 z 2 k k Pochhammer 3 2 k -1 [/itex]

 Rule Form

 Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[SubscriptBox["\[PartialD]", RowBox[List["\[Nu]_"]]], RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]_", ",", "z_"]], "]"]]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[FractionBox["1", "2"], " ", RowBox[List["Log", "[", "2", "]"]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]", ",", "z"]], "]"]]]], "+", RowBox[List[SuperscriptBox["2", RowBox[List["-", FractionBox["\[Nu]", "2"]]]], " ", SuperscriptBox["\[ExponentialE]", RowBox[List["-", FractionBox[SuperscriptBox["z", "2"], "4"]]]], " ", RowBox[List["(", RowBox[List[FractionBox[RowBox[List[RowBox[List[RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]], " ", RowBox[List["PolyGamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]], "-", RowBox[List[SqrtBox["2"], " ", "z", " ", RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", FractionBox["3", "2"], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]]]], RowBox[List["4", " ", RowBox[List["Gamma", "[", RowBox[List["-", "\[Nu]"]], "]"]]]]], "-", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "-", FractionBox["1", "2"]]]], " ", SqrtBox["\[Pi]"], " ", "z"]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", FractionBox["3", "2"], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["PolyGamma", "[", RowBox[List[FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "+", "k"]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["3", "2"], ",", "k"]], "]"]]]]]]]]], ")"]]]], RowBox[List["Gamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]]], "+", FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["\[Nu]", "-", "1"]]], " ", SqrtBox["\[Pi]"]]], ")"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["-", FractionBox["\[Nu]", "2"]]], "]"]], " ", RowBox[List["Hypergeometric1F1", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", FractionBox["1", "2"], ",", FractionBox[SuperscriptBox["z", "2"], "2"]]], "]"]]]], "-", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox["2", RowBox[List["-", "k"]]], " ", SuperscriptBox["z", RowBox[List["2", " ", "k"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", FractionBox["\[Nu]", "2"]]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["k", "-", FractionBox["\[Nu]", "2"]]], "]"]]]], RowBox[List[RowBox[List["k", "!"]], " ", RowBox[List["Pochhammer", "[", RowBox[List[FractionBox["1", "2"], ",", "k"]], "]"]]]]]]]]], ")"]]]], RowBox[List["Gamma", "[", FractionBox[RowBox[List["1", "-", "\[Nu]"]], "2"], "]"]]]]], ")"]]]]]]]]]]

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

 2007-05-02