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http://functions.wolfram.com/06.11.06.0005.02
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LogGamma[z] == LogGamma[Subscript[z, 0]] + PolyGamma[Subscript[z, 0]]
(z - Subscript[z, 0]) + Sum[((-1)^j (z - Subscript[z, 0])^(j + 2))/
((j + 2) (k + Subscript[z, 0])^(j + 2)), {k, 0, Infinity},
{j, 0, Infinity}] /; !(Element[Subscript[z, 0], Reals] &&
Subscript[z, 0] <= 0)
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Cell[BoxData[RowBox[List[RowBox[List[RowBox[List["LogGamma", "[", "z", "]"]], "\[Equal]", RowBox[List[RowBox[List["LogGamma", "[", SubscriptBox["z", "0"], "]"]], "+", RowBox[List[RowBox[List["PolyGamma", "[", SubscriptBox["z", "0"], "]"]], RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["z", "0"]]], ")"]], RowBox[List["j", "+", "2"]]]]], RowBox[List[RowBox[List["(", RowBox[List["j", "+", "2"]], ")"]], SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", SubscriptBox["z", "0"]]], ")"]], RowBox[List["j", "+", "2"]]]]]]]]]]]]]], "/;", RowBox[List["Not", "[", RowBox[List[RowBox[List[SubscriptBox["z", "0"], "\[Element]", "Reals"]], "\[And]", RowBox[List[SubscriptBox["z", "0"], "\[LessEqual]", "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> <mi> logΓ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mo> ⩵ </mo> <mrow> <mrow> <mi> logΓ </mi> <mo> ⁡ </mo> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> + </mo> <mrow> <mrow> <semantics> <mi> ψ </mi> <annotation encoding='Mathematica'> TagBox["\[Psi]", PolyGamma] </annotation> </semantics> <mo> ( </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> j </mi> </msup> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> z </mi> <mo> - </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> j </mi> <mo> + </mo> <mn> 2 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mi> k </mi> <mo> + </mo> <msub> <mi> z </mi> <mn> 0 </mn> </msub> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> + </mo> <mn> 2 </mn> </mrow> </msup> </mrow> </mfrac> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mo> ¬ </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ∈ </mo> <semantics> <mi> ℝ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubleStruckCapitalR]", Function[Integers]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> z </mi> <mn> 0 </mn> </msub> <mo> ≤ </mo> <mn> 0 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <eq /> <apply> <ci> LogGamma </ci> <ci> z </ci> </apply> <apply> <plus /> <apply> <ci> LogGamma </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <times /> <apply> <ci> PolyGamma </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> <apply> <power /> <apply> <plus /> <ci> z </ci> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> <apply> <plus /> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <plus /> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> k </ci> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> <apply> <plus /> <ci> j </ci> <cn type='integer'> 2 </cn> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <not /> <apply> <and /> <apply> <in /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <integers /> </apply> <apply> <leq /> <apply> <ci> Subscript </ci> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>
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| Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["LogGamma", "[", "z_", "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["LogGamma", "[", SubscriptBox["zz", "0"], "]"]], "+", RowBox[List[RowBox[List["PolyGamma", "[", SubscriptBox["zz", "0"], "]"]], " ", RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]]]], "+", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "\[Infinity]"], FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "j"], " ", SuperscriptBox[RowBox[List["(", RowBox[List["z", "-", SubscriptBox["zz", "0"]]], ")"]], RowBox[List["j", "+", "2"]]]]], RowBox[List[RowBox[List["(", RowBox[List["j", "+", "2"]], ")"]], " ", SuperscriptBox[RowBox[List["(", RowBox[List["k", "+", SubscriptBox["zz", "0"]]], ")"]], RowBox[List["j", "+", "2"]]]]]]]]]]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List[SubscriptBox["zz", "0"], "\[Element]", "Reals"]], "&&", RowBox[List[SubscriptBox["zz", "0"], "\[LessEqual]", "0"]]]], ")"]]]]]]]]]] |
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Date Added to functions.wolfram.com (modification date)
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