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http://functions.wolfram.com/01.02.24.0013.01
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Product[k^(1/k^2), {k, 1, Infinity}] == (Glaisher^12/(2 Pi E^EulerGamma))^
(Pi^2/6)
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Cell[BoxData[RowBox[List[RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "\[Infinity]"], SuperscriptBox["k", FractionBox["1", SuperscriptBox["k", "2"]]]]], "\[Equal]", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["Glaisher", "12"], RowBox[List["2", "\[Pi]", " ", SuperscriptBox["\[ExponentialE]", "EulerGamma"]]]], ")"]], FractionBox[SuperscriptBox["\[Pi]", "2"], "6"]]]]]]
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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> <munderover> <mo> ∏ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <msup> <mi> k </mi> <mfrac> <mn> 1 </mn> <msup> <mi> k </mi> <mn> 2 </mn> </msup> </mfrac> </msup> </mrow> <mo> ⩵ </mo> <msup> <mrow> <mo> ( </mo> <mfrac> <msup> <mi> Glaisher </mi> <mn> 12 </mn> </msup> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> π </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[EulerGamma]] </annotation> </semantics> </msup> </mrow> </mfrac> <mo> ) </mo> </mrow> <mfrac> <msup> <mi> π </mi> <mn> 2 </mn> </msup> <mn> 6 </mn> </mfrac> </msup> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <power /> <ci> k </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <power /> <ci> k </ci> <cn type='integer'> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> Glaisher </ci> <cn type='integer'> 12 </cn> </apply> <apply> <power /> <apply> <times /> <cn type='integer'> 2 </cn> <pi /> <apply> <power /> <exponentiale /> <eulergamma /> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k_", "=", "1"]], "\[Infinity]"], SuperscriptBox["k_", FractionBox["1", SuperscriptBox["k_", "2"]]]]], "]"]], "\[RuleDelayed]", SuperscriptBox[RowBox[List["(", FractionBox[SuperscriptBox["Glaisher", "12"], RowBox[List["2", " ", "\[Pi]", " ", SuperscriptBox["\[ExponentialE]", "EulerGamma"]]]], ")"]], FractionBox[SuperscriptBox["\[Pi]", "2"], "6"]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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