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http://functions.wolfram.com/06.11.03.0004.01
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LogGamma[n + 1/4] == LogGamma[1/4] + Sum[Log[4 k - 3], {k, 1, n}] -
2 n Log[2] /; Element[n, Integers] && n >= 0
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LogGamma", "[", RowBox[List["n", "+", FractionBox["1", "4"]]], "]"]], "\[Equal]", RowBox[List[RowBox[List["LogGamma", "[", FractionBox["1", "4"], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List["Log", "[", RowBox[List[RowBox[List["4", "k"]], "-", "3"]], "]"]]]], "-", RowBox[List["2", "n", " ", RowBox[List["Log", "[", "2", "]"]]]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]
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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> <mrow> <mstyle scriptlevel='0'> <mi> logΓ </mi> </mstyle> <mstyle scriptlevel='0'> <mo> ( </mo> </mstyle> <mrow> <mi> n </mi> <mo> + </mo> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mstyle scriptlevel='0'> <mi> logΓ </mi> </mstyle> <mstyle scriptlevel='0'> <mo> ( </mo> </mstyle> <mfrac> <mn> 1 </mn> <mn> 4 </mn> </mfrac> <mo> ) </mo> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> ⁢ </mo> <mi> n </mi> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mn> 4 </mn> <mo> ⁢ </mo> <mi> k </mi> </mrow> <mo> - </mo> <mn> 3 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LogGamma </ci> <apply> <plus /> <ci> n </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> </apply> <apply> <plus /> <apply> <ci> LogGamma </ci> <cn type='rational'> 1 <sep /> 4 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <ci> n </ci> <apply> <ln /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <ln /> <apply> <plus /> <apply> <times /> <cn type='integer'> 4 </cn> <ci> k </ci> </apply> <cn type='integer'> -3 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LogGamma", "[", RowBox[List["n_", "+", FractionBox["1", "4"]]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["LogGamma", "[", FractionBox["1", "4"], "]"]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "1"]], "n"], RowBox[List["Log", "[", RowBox[List[RowBox[List["4", " ", "k"]], "-", "3"]], "]"]]]], "-", RowBox[List["2", " ", "n", " ", RowBox[List["Log", "[", "2", "]"]]]]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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