Wolfram Researchfunctions.wolfram.comOther Wolfram Sites
Search Site
Function CategoriesGraphics GalleryNotationsGeneral IdentitiesAbout This Site ContributeEmail CommentsSign the Guestbook

View Related Information In
The Documentation Center
MathWorld

Download All Formulas For This Function
Mathematica Notebook
PDF File

Download All Introductions For This Function
Mathematica Notebook
PDF File

 

Developed with Mathematica -- Download a Free Trial Version
 











EulerGamma






Mathematica Notation

Traditional Notation









Constants > EulerGamma > Limit representations





http://functions.wolfram.com/02.06.09.0011.01









  


  










Input Form





EulerGamma == Limit[(Subscript[A, n] - Subscript[L, n])/Binomial[2 n, n], n -> Infinity] /; Element[n, Integers] && n >= 1 && Subscript[A, n] == Sum[Binomial[n, i]^2 HarmonicNumber[n + i], {i, 0, n}] && Subscript[L, n] == (1/d[2 n]) Log[Subscript[S, n]] && Subscript[S, n] == Product[(n + k)^(2 (Subscript[d, 2 n]/j) Binomial[n, i]^2), {k, 1, n}, {i, 0, Min[k - 1, n - k]}, {j, i + 1, n - i}] && Subscript[d, n] == LCM[1, 2, \[Ellipsis], n]










Standard Form





Cell[BoxData[RowBox[List[RowBox[List["EulerGamma", "\[Equal]", RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List[SubscriptBox["A", "n"], "-", SubscriptBox["L", "n"]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["2", "n"]], ",", "n"]], "]"]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "1"]], "\[And]", RowBox[List[SubscriptBox["A", "n"], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["Binomial", "[", RowBox[List["n", ",", "i"]], "]"]], "2"], RowBox[List["HarmonicNumber", "[", RowBox[List["n", "+", "i"]], "]"]]]]]]]], "\[And]", "\n", RowBox[List[SubscriptBox["L", "n"], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["d", "[", RowBox[List["2", "n"]], "]"]]], RowBox[List["Log", "[", SubscriptBox["S", "n"], "]"]]]]]], "\[And]", RowBox[List[SubscriptBox["S", "n"], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["i", "=", "0"]], RowBox[List["Min", "[", RowBox[List[RowBox[List["k", "-", "1"]], ",", RowBox[List["n", "-", "k"]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", RowBox[List["i", "+", "1"]]]], RowBox[List["n", "-", "i"]]], SuperscriptBox[RowBox[List["(", RowBox[List["n", "+", "k"]], ")"]], RowBox[List["2", FractionBox[SubscriptBox["d", RowBox[List["2", "n"]]], "j"], SuperscriptBox[RowBox[List["Binomial", "[", RowBox[List["n", ",", "i"]], "]"]], "2"]]]]]]]]]]]], "\[And]", RowBox[List[SubscriptBox["d", "n"], "\[Equal]", RowBox[List["LCM", "[", RowBox[List["1", ",", "2", ",", "\[Ellipsis]", ",", "n"]], "]"]]]]]]]]]]










MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> <mo> &#10869; </mo> <mrow> <munder> <mi> lim </mi> <mrow> <mi> n </mi> <semantics> <mo> &#8594; </mo> <annotation encoding='Mathematica'> &quot;\[Rule]&quot; </annotation> </semantics> <mi> &#8734; </mi> </mrow> </munder> <mo> &#8290; </mo> <mtext> &#8201; </mtext> <mfrac> <mrow> <msub> <mi> A </mi> <mi> n </mi> </msub> <mo> - </mo> <msub> <mi> L </mi> <mi> n </mi> </msub> </mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </mtd> </mtr> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[RowBox[List[&quot;2&quot;, &quot; &quot;, &quot;n&quot;]], Identity, Rule[Editable, True]]], List[TagBox[&quot;n&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> </mfrac> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> &#8712; </mo> <msup> <mi> &#8469; </mi> <mo> + </mo> </msup> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> A </mi> <mi> n </mi> </msub> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <msup> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> i </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;n&quot;, Identity, Rule[Editable, True]]], List[TagBox[&quot;i&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msub> <semantics> <mi> H </mi> <annotation-xml encoding='MathML-Content'> <ci> HarmonicNumber </ci> </annotation-xml> </semantics> <mrow> <mi> i </mi> <mo> + </mo> <mi> n </mi> </mrow> </msub> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> L </mi> <mi> n </mi> </msub> <mo> &#10869; </mo> <mfrac> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <msub> <mi> S </mi> <mi> n </mi> </msub> <mo> ) </mo> </mrow> <mrow> <mi> d </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> S </mi> <mi> n </mi> </msub> <mo> &#10869; </mo> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mrow> <mi> Min </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mi> k </mi> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> , </mo> <mrow> <mi> n </mi> <mo> - </mo> <mi> k </mi> </mrow> </mrow> <mo> ) </mo> </mrow> </munderover> <mrow> <munderover> <mo> &#8719; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mrow> <mi> i </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mrow> <mrow> <mi> n </mi> <mo> - </mo> <mi> i </mi> </mrow> </munderover> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msub> <mi> d </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> n </mi> </mrow> </msub> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> i </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, GridBox[List[List[TagBox[&quot;n&quot;, Identity, Rule[Editable, True]]], List[TagBox[&quot;i&quot;, Identity, Rule[Editable, True]]]]], &quot;)&quot;]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mn> 2 </mn> </msup> <mo> / </mo> <mi> j </mi> </mrow> </mrow> </msup> </mrow> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> d </mi> <mi> n </mi> </msub> <mo> &#10869; </mo> <mrow> <mi> lcm </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> , </mo> <mn> 2 </mn> <mo> , </mo> <mo> &#8230; </mo> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <eulergamma /> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <times /> <apply> <plus /> <apply> <ci> Subscript </ci> <ci> A </ci> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> L </ci> <ci> n </ci> </apply> </apply> </apply> <apply> <power /> <apply> <ci> Binomial </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> <ci> n </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <apply> <ci> SuperPlus </ci> <ci> &#8469; </ci> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> A </ci> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <power /> <apply> <ci> Binomial </ci> <ci> n </ci> <ci> i </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> HarmonicNumber </ci> <apply> <plus /> <ci> i </ci> <ci> n </ci> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> L </ci> <ci> n </ci> </apply> <apply> <times /> <apply> <ln /> <apply> <ci> Subscript </ci> <ci> S </ci> <ci> n </ci> </apply> </apply> <apply> <power /> <apply> <ci> d </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> S </ci> <ci> n </ci> </apply> <apply> <product /> <bvar> <ci> j </ci> </bvar> <lowlimit> <apply> <plus /> <ci> i </ci> <cn type='integer'> 1 </cn> </apply> </lowlimit> <uplimit> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> i </ci> </apply> </apply> </uplimit> <apply> <product /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <apply> <min /> <apply> <plus /> <ci> k </ci> <cn type='integer'> -1 </cn> </apply> <apply> <plus /> <ci> n </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </uplimit> <apply> <product /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <power /> <apply> <plus /> <ci> k </ci> <ci> n </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <ci> Subscript </ci> <ci> d </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> </apply> </apply> <apply> <times /> <apply> <power /> <apply> <ci> Binomial </ci> <ci> n </ci> <ci> i </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> d </ci> <ci> n </ci> </apply> <apply> <lcm /> <cn type='integer'> 1 </cn> <cn type='integer'> 2 </cn> <ci> &#8230; </ci> <ci> n </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", "EulerGamma", "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["Limit", "[", RowBox[List[FractionBox[RowBox[List[SubscriptBox["A", "n"], "-", SubscriptBox["L", "n"]]], RowBox[List["Binomial", "[", RowBox[List[RowBox[List["2", " ", "n"]], ",", "n"]], "]"]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "1"]], "&&", RowBox[List[SubscriptBox["A", "n"], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["i", "=", "0"]], "n"], RowBox[List[SuperscriptBox[RowBox[List["Binomial", "[", RowBox[List["n", ",", "i"]], "]"]], "2"], " ", RowBox[List["HarmonicNumber", "[", RowBox[List["n", "+", "i"]], "]"]]]]]]]], "&&", RowBox[List[SubscriptBox["L", "n"], "\[Equal]", FractionBox[RowBox[List["Log", "[", SubscriptBox["S", "n"], "]"]], RowBox[List["d", "[", RowBox[List["2", " ", "n"]], "]"]]]]], "&&", RowBox[List[SubscriptBox["S", "n"], "\[Equal]", RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["k", "=", "1"]], "n"], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["i", "=", "0"]], RowBox[List["Min", "[", RowBox[List[RowBox[List["k", "-", "1"]], ",", RowBox[List["n", "-", "k"]]]], "]"]]], RowBox[List[UnderoverscriptBox["\[Product]", RowBox[List["j", "=", RowBox[List["i", "+", "1"]]]], RowBox[List["n", "-", "i"]]], SuperscriptBox[RowBox[List["(", RowBox[List["n", "+", "k"]], ")"]], FractionBox[RowBox[List["2", " ", SubscriptBox["d", RowBox[List["2", " ", "n"]]], " ", SuperscriptBox[RowBox[List["Binomial", "[", RowBox[List["n", ",", "i"]], "]"]], "2"]]], "j"]]]]]]]]]], "&&", RowBox[List[SubscriptBox["d", "n"], "\[Equal]", RowBox[List["LCM", "[", RowBox[List["1", ",", "2", ",", "\[Ellipsis]", ",", "n"]], "]"]]]]]]]]]]]]










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





2002-12-18





© 1998- Wolfram Research, Inc.