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variants of this functions
Factorial2






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Factorial2[n] > Series representations > Generalized power series > Expansions at n==n0/;n0!=-2m





http://functions.wolfram.com/06.02.06.0003.02









  


  










Input Form





n!! \[Proportional] Subscript[n, 0]!! (1 + (1/4) (Log[4] + 2 PolyGamma[1 + Subscript[n, 0]/2] + Pi Log[2/Pi] Sin[Subscript[n, 0] Pi]) (n - Subscript[n, 0]) + O[(n - Subscript[n, 0])^2]) /; (n -> Subscript[n, 0]) && !(Element[-(Subscript[n, 0]/2), Integers] && -(Subscript[n, 0]/2) > 0)










Standard Form





Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["n", "!!"]], "\[Proportional]", " ", RowBox[List[RowBox[List[SubscriptBox["n", "0"], "!!"]], RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox["1", "4"], RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "4", "]"]], "+", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", FractionBox[SubscriptBox["n", "0"], "2"]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Log", "[", FractionBox["2", "\[Pi]"], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[SubscriptBox["n", "0"], " ", "\[Pi]"]], "]"]]]]]], ")"]], RowBox[List["(", RowBox[List["n", "-", SubscriptBox["n", "0"]]], ")"]]]], "+", RowBox[List["O", "[", SuperscriptBox[RowBox[List["(", RowBox[List["n", "-", SubscriptBox["n", "0"]]], ")"]], "2"], "]"]]]], ")"]]]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["n", "\[Rule]", SubscriptBox["n", "0"]]], ")"]], "\[And]", RowBox[List["Not", "[", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["n", "0"], "2"]]], "\[Element]", "Integers"]], "\[And]", RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["n", "0"], "2"]]], ">", "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> <mrow> <mi> n </mi> <mo> !! </mo> </mrow> <mo> &#8733; </mo> <mrow> <mrow> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> !! </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 4 </mn> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mfrac> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mn> 2 </mn> </mfrac> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> &#960; </mi> <mo> &#8290; </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mfrac> <mn> 2 </mn> <mi> &#960; </mi> </mfrac> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mi> sin </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mo> &#8290; </mo> <mi> &#960; </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <msub> <mi> n </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <mo> - </mo> <msub> <mi> n </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> n </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <msub> <mi> n </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mo> &#8743; </mo> <mrow> <mrow> <mo> - </mo> <mfrac> <msub> <mi> n </mi> <mn> 0 </mn> </msub> <mn> 2 </mn> </mfrac> </mrow> <mo> &#8713; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Factorial2 </ci> <ci> n </ci> </apply> <apply> <times /> <apply> <ci> Factorial2 </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <cn type='rational'> 1 <sep /> 4 </cn> <apply> <plus /> <apply> <ln /> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> PolyGamma </ci> <apply> <plus /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <pi /> <apply> <ln /> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <sin /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <pi /> </apply> </apply> </apply> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <ci> Rule </ci> <ci> n </ci> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <notin /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Subscript </ci> <ci> n </ci> <cn type='integer'> 0 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["n_", "!!"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List[SubscriptBox["nn", "0"], "!!"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[FractionBox["1", "4"], " ", RowBox[List["(", RowBox[List[RowBox[List["Log", "[", "4", "]"]], "+", RowBox[List["2", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", FractionBox[SubscriptBox["nn", "0"], "2"]]], "]"]]]], "+", RowBox[List["\[Pi]", " ", RowBox[List["Log", "[", FractionBox["2", "\[Pi]"], "]"]], " ", RowBox[List["Sin", "[", RowBox[List[SubscriptBox["nn", "0"], " ", "\[Pi]"]], "]"]]]]]], ")"]], " ", RowBox[List["(", RowBox[List["n", "-", SubscriptBox["nn", "0"]]], ")"]]]], "+", SuperscriptBox[RowBox[List["O", "[", RowBox[List["n", "-", SubscriptBox["nn", "0"]]], "]"]], "2"]]], ")"]]]], "/;", RowBox[List[RowBox[List["(", RowBox[List["n", "\[Rule]", SubscriptBox["nn", "0"]]], ")"]], "&&", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["nn", "0"], "2"]]], "\[Element]", "Integers"]], "&&", RowBox[List[RowBox[List["-", FractionBox[SubscriptBox["nn", "0"], "2"]]], ">", "0"]]]], ")"]]]]]]]]]]]]










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





2001-10-29