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ParabolicCylinderD






Mathematica Notation

Traditional Notation









Hypergeometric Functions > ParabolicCylinderD[nu,z] > Differentiation > Symbolic differentiation > With respect to nu





http://functions.wolfram.com/07.41.20.0013.01









  


  










Input Form





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










Standard Form





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










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mrow> <msup> <mo> &#8706; </mo> <mi> m </mi> </msup> <mrow> <msub> <semantics> <mi> D </mi> <annotation encoding='Mathematica'> TagBox[&quot;D&quot;, ParabolicCylinderD] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mrow> <mo> &#8706; </mo> <msup> <mi> &#957; </mi> <mi> m </mi> </msup> </mrow> </mfrac> <mo> &#63449; </mo> <mrow> <msup> <mn> 2 </mn> <mrow> <mfrac> <mi> &#957; </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mi> m </mi> </mrow> </msup> <mo> &#8290; </mo> <mrow> <msup> <mi> log </mi> <mi> m </mi> </msup> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msqrt> <mi> &#960; </mi> </msqrt> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mo> - </mo> <mfrac> <msup> <mi> z </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> m </mi> </munderover> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> m </mi> </mtd> </mtr> <mtr> <mtd> <mi> k </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;m&quot;, Identity, Rule[Editable, True], Rule[Selectable, True]]], List[TagBox[&quot;k&quot;, Identity, Rule[Editable, True], Rule[Selectable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False], Rule[Selectable, False]] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mfrac> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; 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Rule Form





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










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





2007-05-02





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