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ParabolicCylinderD






Mathematica Notation

Traditional Notation









Hypergeometric Functions > ParabolicCylinderD[nu,z] > Transformations > Addition formulas





http://functions.wolfram.com/07.41.16.0010.01









  


  










Input Form





ParabolicCylinderD[\[Nu], Subscript[z, 1] Cos[\[Alpha]] + Subscript[z, 2] Sin[\[Alpha]]] == Cos[\[Alpha]]^\[Nu] E^((1/4) (Subscript[z, 1] Sin[\[Alpha]] - Subscript[z, 2] Cos[\[Alpha]])^2) Sum[(((-Tan[\[Alpha]])^k Pochhammer[-\[Nu], k])/k!) ParabolicCylinderD[\[Nu] - k, Subscript[z, 1]] ParabolicCylinderD[k, Subscript[z, 2]], {k, 0, Infinity}]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]", ",", RowBox[List[RowBox[List[SubscriptBox["z", "1"], " ", RowBox[List["Cos", "[", "\[Alpha]", "]"]]]], "+", RowBox[List[SubscriptBox["z", "2"], " ", RowBox[List["Sin", "[", "\[Alpha]", "]"]]]]]]]], "]"]], "\[Equal]", RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", "\[Alpha]", "]"]], "\[Nu]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["z", "1"], " ", RowBox[List["Sin", "[", "\[Alpha]", "]"]]]], "-", RowBox[List[SubscriptBox["z", "2"], " ", RowBox[List["Cos", "[", "\[Alpha]", "]"]]]]]], ")"]], "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List["Tan", "[", "\[Alpha]", "]"]]]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], RowBox[List["k", "!"]]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List[RowBox[List["\[Nu]", "-", "k"]], ",", SubscriptBox["z", "1"]]], "]"]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List["k", ",", SubscriptBox["z", "2"]]], "]"]]]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <msub> <semantics> <mi> D </mi> <annotation encoding='Mathematica'> TagBox[&quot;D&quot;, ParabolicCylinderD] </annotation> </semantics> <mi> &#957; </mi> </msub> <mo> ( </mo> <mrow> <mrow> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> </mrow> <mo> + </mo> <mrow> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#63449; </mo> <mrow> <mrow> <msup> <mi> cos </mi> <mi> &#957; </mi> </msup> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#8519; </mi> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> &#8290; </mo> <mrow> <mi> cos </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mrow> <mi> tan </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> &#945; </mi> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mi> &#957; </mi> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List[&quot;(&quot;, RowBox[List[&quot;-&quot;, &quot;\[Nu]&quot;]], &quot;)&quot;]], &quot;k&quot;], Pochhammer] </annotation> </semantics> </mrow> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> D </mi> <annotation encoding='Mathematica'> TagBox[&quot;D&quot;, ParabolicCylinderD] </annotation> </semantics> <mrow> <mi> &#957; </mi> <mo> - </mo> <mi> k </mi> </mrow> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 1 </mn> </msub> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <msub> <semantics> <mi> D </mi> <annotation encoding='Mathematica'> TagBox[&quot;D&quot;, ParabolicCylinderD] </annotation> </semantics> <mi> k </mi> </msub> <mo> ( </mo> <msub> <mi> z </mi> <mn> 2 </mn> </msub> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> ParabolicCylinderD </ci> <ci> &#957; </ci> <apply> <plus /> <apply> <times /> <apply> <cos /> <ci> &#945; </ci> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <times /> <apply> <sin /> <ci> &#945; </ci> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <cos /> <ci> &#945; </ci> </apply> <ci> &#957; </ci> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <power /> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> <apply> <sin /> <ci> &#945; </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <cos /> <ci> &#945; </ci> </apply> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <power /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <tan /> <ci> &#945; </ci> </apply> </apply> <ci> k </ci> </apply> <apply> <ci> Pochhammer </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> &#957; </ci> </apply> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> ParabolicCylinderD </ci> <apply> <plus /> <ci> &#957; </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> ParabolicCylinderD </ci> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["ParabolicCylinderD", "[", RowBox[List["\[Nu]_", ",", RowBox[List[RowBox[List[SubscriptBox["z_", "1"], " ", RowBox[List["Cos", "[", "\[Alpha]_", "]"]]]], "+", RowBox[List[SubscriptBox["z_", "2"], " ", RowBox[List["Sin", "[", "\[Alpha]_", "]"]]]]]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[SuperscriptBox[RowBox[List["Cos", "[", "\[Alpha]", "]"]], "\[Nu]"], " ", SuperscriptBox["\[ExponentialE]", RowBox[List[FractionBox["1", "4"], " ", SuperscriptBox[RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["zz", "1"], " ", RowBox[List["Sin", "[", "\[Alpha]", "]"]]]], "-", RowBox[List[SubscriptBox["zz", "2"], " ", RowBox[List["Cos", "[", "\[Alpha]", "]"]]]]]], ")"]], "2"]]]], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", RowBox[List["Tan", "[", "\[Alpha]", "]"]]]], ")"]], "k"], " ", RowBox[List["Pochhammer", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", "k"]], "]"]]]], ")"]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List[RowBox[List["\[Nu]", "-", "k"]], ",", SubscriptBox["zz", "1"]]], "]"]], " ", RowBox[List["ParabolicCylinderD", "[", RowBox[List["k", ",", SubscriptBox["zz", "2"]]], "]"]]]], RowBox[List["k", "!"]]]]]]]]]]]










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





2007-05-02





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