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http://functions.wolfram.com/07.14.06.0052.01
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GegenbauerC[\[Nu], \[Lambda], z] \[Proportional]
Piecewise[{{0, Element[-\[Nu], Integers] && -\[Nu] > 0},
{ComplexInfinity, Element[-\[Nu] - 2 \[Lambda], Integers] &&
-\[Nu] - 2 \[Lambda] >= 0},
{((-1)^Floor[\[Nu]/2] Pochhammer[\[Lambda], \[Nu] - Floor[\[Nu]/2]]
(2 z)^(\[Nu] - 2 Floor[\[Nu]/2]))/(Floor[\[Nu]/2]!
(\[Nu] - 2 Floor[\[Nu]/2])!), Element[\[Nu], Integers] &&
\[Nu] >= 0}}, (Cos[(Pi \[Nu])/2] Gamma[\[Nu]/2 + \[Lambda]])/
(Gamma[\[Nu]/2 + 1] Gamma[\[Lambda]])] /; (z -> 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]", ",", "\[Lambda]", ",", "z"]], "]"]], "\[Proportional]", RowBox[List["Piecewise", "[", RowBox[List[RowBox[List["{", RowBox[List[RowBox[List["{", RowBox[List["0", ",", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", "\[Nu]"]], ">", "0"]]]]]], "}"]], ",", "\[IndentingNewLine]", RowBox[List["{", RowBox[List["ComplexInfinity", " ", ",", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", "\[Lambda]"]]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", "\[Lambda]"]]]], "\[GreaterEqual]", "0"]]]]]], "}"]], ",", RowBox[List["{", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Lambda]", ",", RowBox[List["\[Nu]", "-", RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List["\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]]]]]]]]], RowBox[List[RowBox[List[RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]]]]]], ")"]], "!"]]]]], ",", RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "\[And]", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]]]], "}"]]]], "}"]], ",", FractionBox[RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Nu]", "2"], "+", "\[Lambda]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]]]], "]"]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]
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<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <msubsup> <mi> C </mi> <mi> ν </mi> <mi> λ </mi> </msubsup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mo>  </mo> <mtable> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mrow> <mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ∈ </mo> <msup> <mi> ℕ </mi> <mo> + </mo> </msup> </mrow> </mtd> </mtr> <mtr> <mtd> <mover> <mi> ∞ </mi> <mo> ~ </mo> </mover> </mtd> <mtd> <mrow> <mrow> <mrow> <mrow> <mo> - </mo> <mn> 2 </mn> </mrow> <mo> ⁢ </mo> <mi> λ </mi> </mrow> <mo> - </mo> <mi> ν </mi> </mrow> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </msup> <mo> ⁢ </mo> <semantics> <msub> <mrow> <mo> ( </mo> <mi> λ </mi> <mo> ) </mo> </mrow> <mrow> <mi> ν </mi> <mo> - </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </msub> <annotation encoding='Mathematica'> TagBox[SubscriptBox[RowBox[List["(", "\[Lambda]", ")"]], RowBox[List["\[Nu]", "-", RowBox[List["\[LeftFloor]", FractionBox["\[Nu]", "2"], "\[RightFloor]"]]]]], Pochhammer] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> z </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> ν </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </mrow> </msup> </mrow> <mrow> <mrow> <mrow> <mo> ⌊ </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> ν </mi> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mrow> <mo> ⌊ </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> <mo> ⌋ </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mrow> </mfrac> </mtd> <mtd> <mrow> <mi> ν </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mfrac> <mrow> <mrow> <mi> cos </mi> <mo> ⁡ </mo> <mo> ( </mo> <mfrac> <mrow> <mi> π </mi> <mo> ⁢ </mo> <mi> ν </mi> </mrow> <mn> 2 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mi> λ </mi> <mo> + </mo> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mi> ν </mi> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> λ </mi> <mo> ) </mo> </mrow> </mrow> </mfrac> </mtd> <mtd> <semantics> <mi> True </mi> <annotation encoding='Mathematica'> TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]] </annotation> </semantics> </mtd> </mtr> </mtable> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mn> 0 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <apply> <power /> <apply> <ci> Subscript </ci> <ci> C </ci> <ci> ν </ci> </apply> <ci> λ </ci> </apply> <ci> z </ci> </apply> <piecewise> <piece> <cn type='integer'> 0 </cn> <apply> <in /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> <apply> <ci> SuperPlus </ci> <ci> ℕ </ci> </apply> </apply> </piece> <piece> <apply> <ci> OverTilde </ci> <infinity /> </apply> <apply> <in /> <apply> <plus /> <apply> <times /> <cn type='integer'> -2 </cn> <ci> λ </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <ci> ν </ci> </apply> </apply> <ci> ℕ </ci> </apply> </piece> <piece> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <ci> Pochhammer </ci> <ci> λ </ci> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <floor /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <apply> <floor /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <factorial /> <apply> <plus /> <ci> ν </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <floor /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <in /> <ci> ν </ci> <ci> ℕ </ci> </apply> </piece> <otherwise> <apply> <times /> <apply> <cos /> <apply> <times /> <pi /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> λ </ci> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <ci> ν </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Gamma </ci> <ci> λ </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </otherwise> </piecewise> </apply> <apply> <ci> Rule </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["GegenbauerC", "[", RowBox[List["\[Nu]_", ",", "\[Lambda]_", ",", "z_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Piecewise]", GridBox[List[List["0", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", "\[Nu]"]], ">", "0"]]]]], List["ComplexInfinity", RowBox[List[RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List[RowBox[List["-", "\[Nu]"]], "-", RowBox[List["2", " ", "\[Lambda]"]]]], "\[GreaterEqual]", "0"]]]]], List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]]], " ", RowBox[List["Pochhammer", "[", RowBox[List["\[Lambda]", ",", RowBox[List["\[Nu]", "-", RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]]]]]], "]"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["2", " ", "z"]], ")"]], RowBox[List["\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]]]]]]]]], RowBox[List[RowBox[List[RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]], "!"]], " ", RowBox[List[RowBox[List["(", RowBox[List["\[Nu]", "-", RowBox[List["2", " ", RowBox[List["Floor", "[", FractionBox["\[Nu]", "2"], "]"]]]]]], ")"]], "!"]]]]], RowBox[List[RowBox[List["\[Nu]", "\[Element]", "Integers"]], "&&", RowBox[List["\[Nu]", "\[GreaterEqual]", "0"]]]]], List[FractionBox[RowBox[List[RowBox[List["Cos", "[", FractionBox[RowBox[List["\[Pi]", " ", "\[Nu]"]], "2"], "]"]], " ", RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Nu]", "2"], "+", "\[Lambda]"]], "]"]]]], RowBox[List[RowBox[List["Gamma", "[", RowBox[List[FractionBox["\[Nu]", "2"], "+", "1"]], "]"]], " ", RowBox[List["Gamma", "[", "\[Lambda]", "]"]]]]], TagBox["True", "PiecewiseDefault", Rule[AutoDelete, False], Rule[DeletionWarning, True]]]], Rule[ColumnAlignments, List[Left]], Rule[ColumnSpacings, 1.2`], Rule[ColumnWidths, Automatic]]]], "/;", RowBox[List["(", RowBox[List["z", "\[Rule]", "0"]], ")"]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
