Gamma[z] == Limit[n^z Beta[z, n], n -> 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> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <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> <mrow> <msup> <mi> n </mi> <mi> z </mi> </msup> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#914; </mi> <annotation-xml encoding='MathML-Content'> <ci> Beta </ci> </annotation-xml> </semantics> <mo> ( </mo> <mrow> <mi> z </mi> <mo> , </mo> <mi> n </mi> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <ci> Gamma </ci> <ci> z </ci> </apply> <apply> <limit /> <bvar> <ci> n </ci> </bvar> <condition> <apply> <tendsto /> <ci> n </ci> <infinity /> </apply> </condition> <apply> <times /> <apply> <power /> <ci> n </ci> <ci> z </ci> </apply> <apply> <ci> Beta </ci> <ci> z </ci> <ci> n </ci> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Gamma", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List["Limit", "[", RowBox[List[RowBox[List[SuperscriptBox["n", "z"], " ", RowBox[List["Beta", "[", RowBox[List["z", ",", "n"]], "]"]]]], ",", RowBox[List["n", "\[Rule]", "\[Infinity]"]]]], "]"]]]]]]

2001-10-29