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http://functions.wolfram.com/02.08.09.0001.01
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Glaisher == Limit[n^(-(n^2/2) - n/2 - 1/12) E^(n^2/4)
Product[k^k, {k, 1, n}], n -> Infinity]
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Cell[BoxData[RowBox[List["Glaisher", "\[Equal]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[SuperscriptBox["n", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["n", "2"], "2"]]], "-", FractionBox["n", "2"], "-", FractionBox["1", "12"]]]], SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["n", "2"], "4"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], SuperscriptBox["k", "k"]]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]]
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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> <semantics> <mi> A </mi> <annotation encoding='Mathematica'> TagBox["A", Function[List[], Glaisher]] </annotation> </semantics> <mo> ⩵ </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> → </mo> <annotation encoding='Mathematica'> "\[Rule]" </annotation> </semantics> <mi> ∞ </mi> </mrow> </munder> <mo> ⁢ </mo> <mtext>   </mtext> <mrow> <msup> <mi> n </mi> <mrow> <mrow> <mo> - </mo> <mfrac> <msup> <mi> n </mi> <mn> 2 </mn> </msup> <mn> 2 </mn> </mfrac> </mrow> <mo> - </mo> <mfrac> <mi> n </mi> <mn> 2 </mn> </mfrac> <mo> - </mo> <mfrac> <mn> 1 </mn> <mn> 12 </mn> </mfrac> </mrow> </msup> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mfrac> <msup> <mi> n </mi> <mn> 2 </mn> </msup> <mn> 4 </mn> </mfrac> </msup> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <msup> <mi> k </mi> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <ci> Glaisher </ci> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <times /> <apply> <power /> <ci> n </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <ci> n </ci> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <cn type='rational'> 1 <sep /> 12 </cn> </apply> </apply> </apply> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <power /> <ci> n </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 4 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <power /> <ci> k </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", "Glaisher", "]"]], "\[RuleDelayed]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[SuperscriptBox["n", RowBox[List[RowBox[List["-", FractionBox[SuperscriptBox["n", "2"], "2"]]], "-", FractionBox["n", "2"], "-", FractionBox["1", "12"]]]], " ", SuperscriptBox["\[ExponentialE]", FractionBox[SuperscriptBox["n", "2"], "4"]], " ", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], SuperscriptBox["k", "k"]]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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