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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 1/Gamma(z) > For the function itself





http://functions.wolfram.com/06.05.06.0040.01









  


  








Input Form





1/Gamma[-n + \[Epsilon]] \[Proportional] \[Epsilon] (-1)^n Sum[Sum[Subscript[a, j] Subscript[b, k - j], {j, 0, k}] \[Epsilon]^k, {k, 0, Infinity}] /; Subscript[a, 2 k] == ((-1)^k Pi^(2 k))/(2 k + 1)! && Subscript[a, 2 k + 1] == 0 && Subscript[b, k] == ((-1)^k/k!) Derivative[k][Gamma][1 + n] && Element[k, Integers] && k >= 0 && Element[n, Integers] && n >= 0








Standard Form





Cell[BoxData[RowBox[List[RowBox[List[FractionBox["1", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n"]], "+", "\[Epsilon]"]], "]"]]], "\[Proportional]", RowBox[List["\[Epsilon]", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SubscriptBox["a", "j"], " ", SubscriptBox["b", RowBox[List["k", "-", "j"]]]]]]], ")"]], " ", SuperscriptBox["\[Epsilon]", "k"]]]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", RowBox[List["2", "k"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], SuperscriptBox["\[Pi]", RowBox[List["2", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", "k"]], "+", "1"]], ")"]], "!"]]]]], "\[And]", RowBox[List[SubscriptBox["a", RowBox[List[RowBox[List["2", "k"]], "+", "1"]]], "\[Equal]", "0"]], "\[And]", RowBox[List[SubscriptBox["b", "k"], "\[Equal]", RowBox[List[FractionBox[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], RowBox[List["k", "!"]]], RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", "+", "n"]], "]"]]]]]], "\[And]", RowBox[List["k", "\[Element]", "Integers"]], "\[And]", RowBox[List["k", "\[GreaterEqual]", "0"]], "\[And]", RowBox[List["n", "\[Element]", "Integers"]], "\[And]", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]








MathML Form







<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mfrac> <mn> 1 </mn> <mrow> <mi> &#915; </mi> <mo> &#8289; </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> + </mo> <mi> &#1013; </mi> </mrow> <mo> ) </mo> </mrow> </mfrac> <mo> &#8733; </mo> <mrow> <mi> &#1013; </mi> <mo> &#8290; </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mo> &#8290; </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> &#8734; </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> &#8721; </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> &#8290; </mo> <msub> <mi> b </mi> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> </mrow> </msub> </mrow> </mrow> <mo> ) </mo> </mrow> <mo> &#8290; </mo> <msup> <mi> &#1013; </mi> <mi> k </mi> </msup> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <msub> <mi> a </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msub> <mo> &#63449; </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mo> &#8290; </mo> <msup> <mi> &#960; </mi> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> </msup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> a </mi> <mrow> <mrow> <mn> 2 </mn> <mo> &#8290; </mo> <mi> k </mi> </mrow> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> <mo> &#63449; </mo> <mn> 0 </mn> </mrow> <mo> &#8743; </mo> <mrow> <msub> <mi> b </mi> <mi> k </mi> </msub> <mo> &#63449; </mo> <mrow> <mfrac> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> k </mi> </msup> <mrow> <mi> k </mi> <mo> ! </mo> </mrow> </mfrac> <mo> &#8290; </mo> <mrow> <msup> <mi> &#915; </mi> <semantics> <mrow> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List[&quot;(&quot;, &quot;k&quot;, &quot;)&quot;]], Derivative] </annotation> </semantics> </msup> <mo> ( </mo> <mrow> <mi> n </mi> <mo> + </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <mo> &#8743; </mo> <mrow> <mi> k </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> <mo> &#8743; </mo> <mrow> <mi> n </mi> <mo> &#8712; </mo> <mi> &#8469; </mi> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Condition </ci> <apply> <ci> Proportional </ci> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <ci> Gamma </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <ci> &#1013; </ci> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <ci> &#1013; </ci> <apply> <power /> <cn type='integer'> -1 </cn> <ci> n </ci> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> b </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> <apply> <power /> <ci> &#1013; </ci> <ci> k </ci> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <pi /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <apply> <plus /> <apply> <times /> <cn type='integer'> 2 </cn> <ci> k </ci> </apply> <cn type='integer'> 1 </cn> </apply> </apply> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> b </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> <apply> <power /> <apply> <factorial /> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <ci> D </ci> <apply> <ci> Gamma </ci> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> </apply> <list> <apply> <plus /> <ci> n </ci> <cn type='integer'> 1 </cn> </apply> <ci> k </ci> </list> </apply> </apply> </apply> <apply> <in /> <ci> k </ci> <ci> &#8469; </ci> </apply> <apply> <in /> <ci> n </ci> <ci> &#8469; </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>










Rule Form





Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", FractionBox["1", RowBox[List["Gamma", "[", RowBox[List[RowBox[List["-", "n_"]], "+", "\[Epsilon]_"]], "]"]]], "]"]], "\[RuleDelayed]", RowBox[List[RowBox[List["\[Epsilon]", " ", SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "n"], " ", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["k", "=", "0"]], "\[Infinity]"], RowBox[List[RowBox[List["(", RowBox[List[UnderoverscriptBox["\[Sum]", RowBox[List["j", "=", "0"]], "k"], RowBox[List[SubscriptBox["a", "j"], " ", SubscriptBox["b", RowBox[List["k", "-", "j"]]]]]]], ")"]], " ", SuperscriptBox["\[Epsilon]", "k"]]]]]]], "/;", RowBox[List[RowBox[List[SubscriptBox["a", RowBox[List["2", " ", "k"]]], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", SuperscriptBox["\[Pi]", RowBox[List["2", " ", "k"]]]]], RowBox[List[RowBox[List["(", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]], ")"]], "!"]]]]], "&&", RowBox[List[SubscriptBox["a", RowBox[List[RowBox[List["2", " ", "k"]], "+", "1"]]], "\[Equal]", "0"]], "&&", RowBox[List[SubscriptBox["b", "k"], "\[Equal]", FractionBox[RowBox[List[SuperscriptBox[RowBox[List["(", RowBox[List["-", "1"]], ")"]], "k"], " ", RowBox[List[SuperscriptBox["Gamma", TagBox[RowBox[List["(", "k", ")"]], Derivative], Rule[MultilineFunction, None]], "[", RowBox[List["1", "+", "n"]], "]"]]]], RowBox[List["k", "!"]]]]], "&&", RowBox[List["k", "\[Element]", "Integers"]], "&&", RowBox[List["k", "\[GreaterEqual]", "0"]], "&&", RowBox[List["n", "\[Element]", "Integers"]], "&&", RowBox[List["n", "\[GreaterEqual]", "0"]]]]]]]]]]


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





2007-05-02