LogGamma[z] \[Proportional] -Log[z] - EulerGamma z + (Pi^2 z^2)/12 - (Zeta[3]/3) z^3 + (Pi^4 z^4)/360 - O[z^5]

Cell[BoxData[RowBox[List[RowBox[List["LogGamma", "[", "z", "]"]], "\[Proportional]", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["EulerGamma", " ", "z"]], "+", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "2"]]], "12"], "-", RowBox[List[FractionBox[RowBox[List["Zeta", "[", "3", "]"]], "3"], " ", SuperscriptBox["z", "3"]]], " ", "+", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "4"], " ", SuperscriptBox["z", "4"]]], "360"], "-", RowBox[List["O", "[", SuperscriptBox["z", "5"], "]"]]]]]]]]

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mstyle scriptlevel='0'> <mrow> <mi> log&#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mstyle> <mo> &#8733; </mo> <mrow> <mrow> <mo> - </mo> <mrow> <mi> log </mi> <mo> &#8289; </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <semantics> <mi> &#8509; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubledGamma]&quot;, Function[EulerGamma]] </annotation> </semantics> <mo> &#8290; </mo> <mi> z </mi> </mrow> <mo> + </mo> <mfrac> <mrow> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 2 </mn> </msup> </mrow> <mn> 12 </mn> </mfrac> <mo> - </mo> <mfrac> <mrow> <mtext> </mtext> <mrow> <semantics> <mrow> <mi> &#950; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;\[Zeta]&quot;, &quot;(&quot;, TagBox[&quot;3&quot;, Rule[Editable, True]], &quot;)&quot;]], InterpretTemplate[Function[$CellContext`e, Zeta[$CellContext`e]]]] </annotation> </semantics> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 3 </mn> </msup> </mrow> </mrow> <mn> 3 </mn> </mfrac> <mo> + </mo> <mfrac> <mrow> <msup> <mi> &#960; </mi> <mn> 4 </mn> </msup> <mo> &#8290; </mo> <msup> <mi> z </mi> <mn> 4 </mn> </msup> </mrow> <mn> 360 </mn> </mfrac> <mo> - </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> z </mi> <mn> 5 </mn> </msup> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Proportional </ci> <apply> <times /> <ci> log&#915; </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ln /> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <eulergamma /> <ci> z </ci> </apply> </apply> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <cn type='integer'> 12 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <ci> Zeta </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 3 </cn> </apply> <apply> <power /> <cn type='integer'> 3 </cn> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <ci> z </ci> <cn type='integer'> 4 </cn> </apply> <apply> <power /> <cn type='integer'> 360 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> O </ci> <apply> <power /> <ci> z </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LogGamma", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["-", RowBox[List["Log", "[", "z", "]"]]]], "-", RowBox[List["EulerGamma", " ", "z"]], "+", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "2"], " ", SuperscriptBox["z", "2"]]], "12"], "-", RowBox[List[FractionBox["1", "3"], " ", RowBox[List["Zeta", "[", "3", "]"]], " ", SuperscriptBox["z", "3"]]], "+", FractionBox[RowBox[List[SuperscriptBox["\[Pi]", "4"], " ", SuperscriptBox["z", "4"]]], "360"], "-", SuperscriptBox[RowBox[List["O", "[", "z", "]"]], "5"]]]]]]]

2007-05-02