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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(z+Epsilon) at Epsilon==0/;z!=-n > For the function itself





http://functions.wolfram.com/06.05.06.0024.01









  


  










Input Form





Gamma[z + \[Epsilon]] \[Proportional] Gamma[z] (1 + PolyGamma[z] \[Epsilon] + (1/2) (PolyGamma[1, z] + PolyGamma[z]^2) \[Epsilon]^2 + (1/6) (PolyGamma[z]^3 + 3 PolyGamma[1, z] PolyGamma[z] + PolyGamma[2, z]) \[Epsilon]^3 + (1/24) (PolyGamma[z]^4 + 6 PolyGamma[z]^2 PolyGamma[1, z] + 3 PolyGamma[1, z]^2 + 4 PolyGamma[z] PolyGamma[2, z] + PolyGamma[3, z]) \[Epsilon]^4 + O[\[Epsilon]^5]) /; !(Element[z, Integers] && z <= 0)










Standard Form





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MathML Form







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</mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mrow> <mo> ( </mo> <mn> 1 </mn> <mo> ) </mo> </mrow> </msup> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mn> 2 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#1013; </mi> <mn> 2 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 3 </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> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <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> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mn> 6 </mn> </mfrac> <mo> &#8290; </mo> <msup> <mi> &#1013; </mi> <mn> 3 </mn> </msup> </mrow> <mo> + </mo> <mrow> <mfrac> <mn> 1 </mn> <mn> 24 </mn> </mfrac> <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> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 4 </mn> </msup> <mo> + </mo> <mrow> <mn> 6 </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> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <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> <mi> z </mi> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <mrow> <semantics> <mi> &#968; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[Psi]&quot;, PolyGamma] </annotation> </semantics> <mo> ( </mo> <mi> z </mi> <mo> ) </mo> </mrow> </mrow> <mo> + </mo> <mrow> <mn> 3 </mn> <mo> &#8290; </mo> <msup> <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> <mi> z </mi> <mo> ) </mo> </mrow> <mn> 2 </mn> </msup> </mrow> <mo> + </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> <mi> z </mi> <mo> ) </mo> </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> <mo> &#172; </mo> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> z </mi> <mo> &#8712; </mo> <semantics> <mi> &#8484; </mi> <annotation encoding='Mathematica'> TagBox[&quot;\[DoubleStruckCapitalZ]&quot;, Function[Integers]] </annotation> </semantics> </mrow> <mo> &#8743; </mo> <mrow> <mi> z </mi> <mo> &#8804; </mo> <mn> 0 </mn> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> z </ci> <ci> &#1013; </ci> </apply> </apply> <apply> <times /> <apply> <ci> Gamma </ci> <ci> z </ci> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <times /> <apply> <ci> PolyGamma </ci> <ci> z </ci> </apply> <ci> &#1013; </ci> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <ci> PolyGamma </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 2 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> &#1013; </ci> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <apply> <times /> <apply> <plus /> <apply> <power /> <apply> <ci> PolyGamma </ci> <ci> z </ci> </apply> <cn type='integer'> 3 </cn> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <apply> <ci> PolyGamma </ci> <ci> z </ci> </apply> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> </apply> <apply> <power /> <cn type='integer'> 6 </cn> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <ci> &#1013; </ci> <cn type='integer'> 3 </cn> </apply> </apply> <apply> <times /> <cn type='rational'> 1 <sep /> 24 </cn> <apply> <plus /> <apply> <power /> <apply> <ci> PolyGamma </ci> <ci> z </ci> </apply> <cn type='integer'> 4 </cn> </apply> <apply> <times /> <cn type='integer'> 6 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <apply> <power /> <apply> <ci> PolyGamma </ci> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> 4 </cn> <apply> <ci> PolyGamma </ci> <cn type='integer'> 2 </cn> <ci> z </ci> </apply> <apply> <ci> PolyGamma </ci> <ci> z </ci> </apply> </apply> <apply> <times /> <cn type='integer'> 3 </cn> <apply> <power /> <apply> <ci> PolyGamma </ci> <cn type='integer'> 1 </cn> <ci> z </ci> </apply> <cn type='integer'> 2 </cn> </apply> </apply> <apply> <ci> PolyGamma </ci> <cn type='integer'> 3 </cn> <ci> z </ci> </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> <not /> <apply> <and /> <apply> <in /> <ci> z </ci> <integers /> </apply> <apply> <leq /> <ci> z </ci> <cn type='integer'> 0 </cn> </apply> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["Gamma", "[", RowBox[List["z_", "+", "\[Epsilon]_"]], "]"]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List[RowBox[List["Gamma", "[", "z", "]"]], " ", RowBox[List["(", RowBox[List["1", "+", RowBox[List[RowBox[List["PolyGamma", "[", "z", "]"]], " ", "\[Epsilon]"]], "+", RowBox[List[FractionBox["1", "2"], " ", RowBox[List["(", RowBox[List[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", "z"]], "]"]], "+", SuperscriptBox[RowBox[List["PolyGamma", "[", "z", "]"]], "2"]]], ")"]], " ", SuperscriptBox["\[Epsilon]", "2"]]], "+", RowBox[List[FractionBox["1", "6"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", "z", "]"]], "3"], "+", RowBox[List["3", " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", "z"]], "]"]], " ", RowBox[List["PolyGamma", "[", "z", "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", "z"]], "]"]]]], ")"]], " ", SuperscriptBox["\[Epsilon]", "3"]]], "+", RowBox[List[FractionBox["1", "24"], " ", RowBox[List["(", RowBox[List[SuperscriptBox[RowBox[List["PolyGamma", "[", "z", "]"]], "4"], "+", RowBox[List["6", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", "z", "]"]], "2"], " ", RowBox[List["PolyGamma", "[", RowBox[List["1", ",", "z"]], "]"]]]], "+", RowBox[List["3", " ", SuperscriptBox[RowBox[List["PolyGamma", "[", RowBox[List["1", ",", "z"]], "]"]], "2"]]], "+", RowBox[List["4", " ", RowBox[List["PolyGamma", "[", "z", "]"]], " ", RowBox[List["PolyGamma", "[", RowBox[List["2", ",", "z"]], "]"]]]], "+", RowBox[List["PolyGamma", "[", RowBox[List["3", ",", "z"]], "]"]]]], ")"]], " ", SuperscriptBox["\[Epsilon]", "4"]]], "+", SuperscriptBox[RowBox[List["O", "[", "\[Epsilon]", "]"]], "5"]]], ")"]]]], "/;", RowBox[List["!", RowBox[List["(", RowBox[List[RowBox[List["z", "\[Element]", "Integers"]], "&&", RowBox[List["z", "\[LessEqual]", "0"]]]], ")"]]]]]]]]]]










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





2007-05-02





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