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variants of this functions
Gamma






Mathematica Notation

Traditional Notation









Gamma, Beta, Erf > Gamma[z] > Series representations > Generalized power series > Expansions of Gamma(-n+Epsilon) at Epsilon==0 > For the function itself





http://functions.wolfram.com/06.05.06.0030.01









  


  










Input Form





Gamma[-n + \[Epsilon]] \[Proportional] ((-1)^n/(n! \[Epsilon])) (1 + PolyGamma[n + 1] \[Epsilon] + ((Pi^2 + 3 PolyGamma[1 + n]^2 - 3 PolyGamma[1, 1 + n])/6) \[Epsilon]^2 + (1/6) (PolyGamma[1 + n]^3 + PolyGamma[1 + n] (Pi^2 - 3 PolyGamma[1, 1 + n]) + PolyGamma[2, 1 + n]) \[Epsilon]^3 + (1/360) (7 Pi^4 + 15 (PolyGamma[1 + n]^4 + 2 PolyGamma[1 + n]^2 (Pi^2 - 3 PolyGamma[1, 1 + n]) + PolyGamma[1, 1 + n] (-2 Pi^2 + 3 PolyGamma[1, 1 + n]) + 4 PolyGamma[1 + n] PolyGamma[2, 1 + n]) - 15 PolyGamma[3, 1 + n]) \[Epsilon]^4 + O[\[Epsilon]^5]) /; Element[n, Integers] && n >= 0










Standard Form





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MathML Form







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</mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#1013; </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 360 </mn> </mfrac> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> <mo> - </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mn> 4 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 2 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> - </mo> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 2 </mn> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 7 </mn> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mn> 4 </mn> </msup> </mrow> <mo> - </mo> <mrow> <mn> 15 </mn> <mo> &#8290; </mo> <mrow> <msup> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 3 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#1013; </mi> <mn> 4 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mi> O </mi> <mo> &#8289; </mo> <mo> ( </mo> <msup> <mi> &#1013; </mi> <mn> 5 </mn> </msup> <mo> ) </mo> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> &#1013; </ci> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <power /> <apply> <times /> <apply> <factorial /> <ci> n </ci> </apply> <ci> &#1013; </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <ci> &#1013; </ci> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> &#1013; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 6 </cn> <apply> <plus /> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <apply> <plus /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> &#1013; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 360 </cn> <apply> <plus /> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <plus /> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <plus /> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> PolyGamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <plus /> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <pi /> <cn type='integer'> 2 </cn> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <times /> <cn type='integer'> 7 </cn> <apply> <power /> <pi /> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <cn type='integer'> 15 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 3 </cn> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> &#1013; </ci> <cn type='integer'> 4 </cn> </apply> </apply> <apply> <ci> O </ci> <apply> <power /> <ci> &#1013; </ci> <cn type='integer'> 5 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n_"]], "+", "\[Epsilon]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["n", "+", "1"]], "]"]], " ", "\[Epsilon]"]], "+", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "n"]], "]"]], "2"]]], "-", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "n"]]]], "]"]]]]]], ")"]], " ", SuperscriptBox["\[Epsilon]", "2"]]], "+", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "n"]], "]"]], "3"], "+", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "n"]], "]"]], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "-", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "n"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", RowBox[List["1", "+", "n"]]]], "]"]]]], ")"]], " ", SuperscriptBox["\[Epsilon]", "3"]]], "+", RowBox[List[FractionBox["1", "360"], " ", RowBox[List["(", RowBox[List[RowBox[List["7", " ", SuperscriptBox["\[Pi]", "4"]]], "+", RowBox[List["15", " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "n"]], "]"]], "4"], "+", RowBox[List["2", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "n"]], "]"]], "2"], " ", RowBox[List["(", RowBox[List[SuperscriptBox["\[Pi]", "2"], "-", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "n"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "n"]]]], "]"]], " ", RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "2"]], " ", SuperscriptBox["\[Pi]", "2"]]], "+", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", RowBox[List["1", "+", "n"]]]], "]"]]]]]], ")"]]]], "+", RowBox[List["4", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", "+", "n"]], "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", RowBox[List["1", "+", "n"]]]], "]"]]]]]], ")"]]]], "-", RowBox[List["15", " ", RowBox[List["PolyGamma", "[", RowBox[List["3", ",", RowBox[List["1", "+", "n"]]]], "]"]]]]]], ")"]], " ", SuperscriptBox["\[Epsilon]", "4"]]], "+", SuperscriptBox[RowBox[List["O", "[", "\[Epsilon]", "]"]], "5"]]], ")"]]]], RowBox[List[RowBox[List["n", "!"]], " ", "\[Epsilon]"]]], "/;", RowBox[List[RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]










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





2007-05-02





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