|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
http://functions.wolfram.com/07.03.04.0006.01
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Residue[LaguerreL[\[Nu], \[Lambda], z], {\[Lambda], -\[Nu] - k}] ==
((-1)^(k - 1)/(Gamma[\[Nu] + 1] (k - 1)!)) Hypergeometric1F1Regularized[
-\[Nu], 1 - \[Nu] - k, z] /; Element[k, Integers] && k > 0
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["Residue", "[", RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]], ",", RowBox[List["{", RowBox[List["\[Lambda]", ",", RowBox[List[RowBox[List["-", "\[Nu]"]], "-", "k"]]]], "}"]]]], "]"]], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], RowBox[List[RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]], "!"]]]]], " ", RowBox[List["Hypergeometric1F1Regularized", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "-", "\[Nu]", "-", "k"]], ",", "z"]], "]"]]]]]], "/;", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", ">", "0"]]]]]]]]
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <msub> <mi> res </mi> <mi> λ </mi> </msub> <mo> ( </mo> <mrow> <msubsup> <mi> L </mi> <mi> ν </mi> <mi> λ </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> <mo> ⁢ </mo> <semantics> <mrow> <mrow> <msub> <mo>   </mo> <mn> 1 </mn> </msub> <msub> <mover> <mi> F </mi> <mo> ~ </mo> </mover> <mn> 1 </mn> </msub> </mrow> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ; </mo> <mrow> <mrow> <mo> - </mo> <mi> k </mi> </mrow> <mo> - </mo> <mi> ν </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ; </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation encoding='Mathematica'> TagBox[TagBox[RowBox[List[RowBox[List[SubscriptBox["\[InvisiblePrefixScriptBase]", FormBox["1", TraditionalForm]], SubscriptBox[OverscriptBox["F", "~"], FormBox["1", TraditionalForm]]]], "\[InvisibleApplication]", RowBox[List["(", RowBox[List[TagBox[TagBox[TagBox[RowBox[List["-", "\[Nu]"]], Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], ";", TagBox[TagBox[TagBox[RowBox[List[RowBox[List["-", "k"]], "-", "\[Nu]", "+", "1"]], Hypergeometric1F1Regularized, Rule[Editable, True]], InterpretTemplate[Function[List[SlotSequence[1]]]]], Hypergeometric1F1Regularized, Rule[Editable, False]], ";", TagBox["z", Hypergeometric1F1Regularized, Rule[Editable, True]]]], ")"]]]], InterpretTemplate[Function[HypergeometricPFQRegularized[Slot[1], Slot[2], Slot[3]]]], Rule[Editable, False]], Hypergeometric1F1Regularized] </annotation> </semantics> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> k </mi> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <times /> <apply> <apply> <ci> Subscript </ci> <ci> res </ci> <ci> λ </ci> </apply> <apply> <ci> LaguerreL </ci> <ci> ν </ci> <ci> λ </ci> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> ν </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> Hypergeometric1F1Regularized </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <cn type='integer'> 1 </cn> </apply> <ci> z </ci> </apply> </apply> </apply> <apply> <in /> <ci> k </ci> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
|
|
|
|
|
|
|
|
|
|
| |
|
|
|
|
| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Residue", "[", RowBox[List[RowBox[List["LaguerreL", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", "z_"]], "]"]], ",", RowBox[List["{", RowBox[List["\[Lambda]_", ",", RowBox[List[RowBox[List["-", "\[Nu]_"]], "-", "k_"]]]], "}"]]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["k", "-", "1"]]], " ", RowBox[List["Hypergeometric1F1Regularized", "[", RowBox[List[RowBox[List["-", "\[Nu]"]], ",", RowBox[List["1", "-", "\[Nu]", "-", "k"]], ",", "z"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List["\[Nu]", "+", "1"]], "]"]], " ", RowBox[List[RowBox[List["(", RowBox[List["k", "-", "1"]], ")"]], "!"]]]]], "/;", RowBox[List[RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", ">", "0"]]]]]]]]]] |
|
|
|
|
|
|
|
|
|
|
Date Added to functions.wolfram.com (modification date)
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|