Gamma function: Series representations (formula 06.05.06.0032)

Gamma

$Mathematica Notation$

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/06.05.06.0032.01

Input Form

Gamma[-n + \[Epsilon]] \[Proportional] ((-1)^n/(n! \[Epsilon])) Sum[((Subscript[p, k] \[Epsilon]^q)/(q - k)!) (1 + Sum[Sum[(Binomial[n, j] (-1)^(j - 1))/j^i, {j, 1, n}] \[Epsilon]^i, {i, 1, Infinity}]), {q, 0, Infinity}, {k, 0, q}] /; Element[n, Integers] && n >= 0 && Subscript[s, 1] == EulerGamma && Subscript[s, k] == Zeta[k] /; k > 1 && Subscript[a, 0] == -EulerGamma && Subscript[a, k] == ((-1)^(k + 1)/(k + 1)) Subscript[s, k + 1] /; k > 0 && Subscript[p, 0] == (-1)^m EulerGamma^m && Subscript[p, k] == (1/(Subscript[a, 0] k)) Sum[(m j - k + j) Subscript[a, j] Subscript[p, k - j], {j, 1, k}] /; k > 0 && m = q - k

Standard Form

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MathML Form

<math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'> <semantics> <mrow> <mrow> <mrow> <mrow> <mrow> <mrow> <mrow> <mi> Γ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mrow> <mo> - </mo> <mi> n </mi> </mrow> <mo> + </mo> <mi> ϵ </mi> </mrow> <mo> ) </mo> </mrow> <mo> ∝ </mo> <mrow> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> n </mi> </msup> <mtext> </mtext> </mrow> <mrow> <mrow> <mi> n </mi> <mo> ! </mo> </mrow> <mo> ⁢ </mo> <mi> ϵ </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> q </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> k </mi> <mo> = </mo> <mn> 0 </mn> </mrow> <mi> q </mi> </munderover> <mrow> <mfrac> <mrow> <msub> <mi> p </mi> <mi> k </mi> </msub> <mo> ⁢ </mo> <msup> <mi> ϵ </mi> <mi> q </mi> </msup> </mrow> <mrow> <mrow> <mo> ( </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ! </mo> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mn> 1 </mn> <mo> + </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> i </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> ∞ </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> n </mi> </munderover> <mfrac> <mrow> <semantics> <mrow> <mo> ( </mo> <mtable> <mtr> <mtd> <mi> n </mi> </mtd> </mtr> <mtr> <mtd> <mi> j </mi> </mtd> </mtr> </mtable> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["(", GridBox[List[List[TagBox["n", Identity, Rule[Editable, True]]], List[TagBox["j", Identity, Rule[Editable, True]]]]], ")"]], InterpretTemplate[Function[Binomial[Slot[1], Slot[2]]]], Rule[Editable, False]] </annotation> </semantics> <mo> ⁢ </mo> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> j </mi> <mo> - </mo> <mn> 1 </mn> </mrow> </msup> </mrow> <msup> <mi> j </mi> <mi> i </mi> </msup> </mfrac> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msup> <mi> ϵ </mi> <mi> i </mi> </msup> </mrow> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> n </mi> <mo> ∈ </mo> <mi> ℕ </mi> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> s </mi> <mn> 1 </mn> </msub> <mo>  </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> s </mi> <mi> k </mi> </msub> <mo>  </mo> <semantics> <mrow> <mi> ζ </mi> <mo> ⁡ </mo> <mo> ( </mo> <mi> k </mi> <mo> ) </mo> </mrow> <annotation encoding='Mathematica'> TagBox[RowBox[List["\[Zeta]", "(", TagBox["k", Zeta, Rule[Editable, True]], ")"]], InterpretTemplate[Function[BoxForm`e$, Zeta[BoxForm`e$]]]] </annotation> </semantics> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> > </mo> <mn> 1 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo>  </mo> <mrow> <mo> - </mo> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> </mrow> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> a </mi> <mi> k </mi> </msub> <mo>  </mo> <mfrac> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msup> <mo> ⁢ </mo> <msub> <mi> s </mi> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </msub> </mrow> <mrow> <mi> k </mi> <mo> + </mo> <mn> 1 </mn> </mrow> </mfrac> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> p </mi> <mn> 0 </mn> </msub> <mo>  </mo> <mrow> <msup> <mrow> <mo> ( </mo> <mrow> <mo> - </mo> <mn> 1 </mn> </mrow> <mo> ) </mo> </mrow> <mi> m </mi> </msup> <mo> ⁢ </mo> <msup> <semantics> <mi> ℽ </mi> <annotation encoding='Mathematica'> TagBox["\[DoubledGamma]", Function[List[], EulerGamma]] </annotation> </semantics> <mi> m </mi> </msup> </mrow> </mrow> <mo> ∧ </mo> <mrow> <msub> <mi> p </mi> <mi> k </mi> </msub> <mo>  </mo> <mrow> <mfrac> <mn> 1 </mn> <mrow> <msub> <mi> a </mi> <mn> 0 </mn> </msub> <mo> ⁢ </mo> <mi> k </mi> </mrow> </mfrac> <mo> ⁢ </mo> <mrow> <munderover> <mo> ∑ </mo> <mrow> <mi> j </mi> <mo> = </mo> <mn> 1 </mn> </mrow> <mi> k </mi> </munderover> <mrow> <mrow> <mo> ( </mo> <mrow> <mrow> <mi> m </mi> <mo> ⁢ </mo> <mi> j </mi> </mrow> <mo> + </mo> <mi> j </mi> <mo> - </mo> <mi> k </mi> </mrow> <mo> ) </mo> </mrow> <mo> ⁢ </mo> <msub> <mi> a </mi> <mi> j </mi> </msub> <mo> ⁢ </mo> <msub> <mi> p </mi> <mrow> <mi> k </mi> <mo> - </mo> <mi> j </mi> </mrow> </msub> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo> /; </mo> <mrow> <mrow> <mi> k </mi> <mo> > </mo> <mn> 0 </mn> </mrow> <mo> ∧ </mo> <mi> m </mi> </mrow> </mrow> <mo> = </mo> <mrow> <mi> q </mi> <mo> - </mo> <mi> k </mi> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <ci> Set </ci> <apply> <ci> Condition </ci> <apply> <ci> Condition </ci> <apply> <ci> Condition </ci> <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> ϵ </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> ϵ </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> k </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <ci> q </ci> </uplimit> <apply> <sum /> <bvar> <ci> q </ci> </bvar> <lowlimit> <cn type='integer'> 0 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> <apply> <power /> <ci> ϵ </ci> <ci> q </ci> </apply> <apply> <power /> <apply> <factorial /> <apply> <plus /> <ci> q </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <plus /> <cn type='integer'> 1 </cn> <apply> <sum /> <bvar> <ci> i </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <infinity /> </uplimit> <apply> <times /> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> n </ci> </uplimit> <apply> <times /> <apply> <ci> Binomial </ci> <ci> n </ci> <ci> j </ci> </apply> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> j </ci> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <power /> <apply> <power /> <ci> j </ci> <ci> i </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <ci> ϵ </ci> <ci> i </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <in /> <ci> n </ci> <ci> ℕ </ci> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> s </ci> <cn type='integer'> 1 </cn> </apply> <eulergamma /> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> s </ci> <ci> k </ci> </apply> <apply> <ci> Zeta </ci> <ci> k </ci> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <eulergamma /> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <ci> Subscript </ci> <ci> s </ci> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> </apply> <apply> <power /> <apply> <plus /> <ci> k </ci> <cn type='integer'> 1 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> p </ci> <cn type='integer'> 0 </cn> </apply> <apply> <times /> <apply> <power /> <cn type='integer'> -1 </cn> <ci> m </ci> </apply> <apply> <power /> <eulergamma /> <ci> m </ci> </apply> </apply> </apply> <apply> <eq /> <apply> <ci> Subscript </ci> <ci> p </ci> <ci> k </ci> </apply> <apply> <times /> <apply> <times /> <cn type='integer'> 1 </cn> <apply> <power /> <apply> <times /> <apply> <ci> Subscript </ci> <ci> a </ci> <cn type='integer'> 0 </cn> </apply> <ci> k </ci> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <sum /> <bvar> <ci> j </ci> </bvar> <lowlimit> <cn type='integer'> 1 </cn> </lowlimit> <uplimit> <ci> k </ci> </uplimit> <apply> <times /> <apply> <plus /> <apply> <times /> <ci> m </ci> <ci> j </ci> </apply> <ci> j </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> <apply> <ci> Subscript </ci> <ci> a </ci> <ci> j </ci> </apply> <apply> <ci> Subscript </ci> <ci> p </ci> <apply> <plus /> <ci> k </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> j </ci> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> </apply> <apply> <and /> <apply> <gt /> <ci> k </ci> <cn type='integer'> 0 </cn> </apply> <ci> m </ci> </apply> </apply> <apply> <plus /> <ci> q </ci> <apply> <times /> <cn type='integer'> -1 </cn> <ci> k </ci> </apply> </apply> </apply> </annotation-xml> </semantics> </math>

Rule Form

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

2007-05-02

Gamma[a,z]
Gamma[a,z₁,z₂]