Some limit formulas for the Gamma and Psi (or Digamma) functions at their singularities

被引:9
|
作者
Prabhu, Anirudh [2 ]
Srivastava, H. M. [1 ]
机构
[1] Univ Victoria, Dept Math & Stat, Victoria, BC V8W 3R4, Canada
[2] W Lafayette Junior Senior High Sch, W Lafayette, IN 47906 USA
关键词
Gamma function; Psi (or Digamma) function; meromorphic continuation; Eulerian integral; singularities and simple poles; limit formulas; L'Hopital's rule; Euler's reflection formula; Gauss-Legendre multiplication formula; Stirling formula;
D O I
10.1080/10652469.2010.535970
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Both the Gamma function Gamma( z) and the Psi (or Digamma) function Psi(z) are meromorphically continued to the whole complex z-plane with singularities (which are simple poles) at z = -k for non-positive integer values of k. Here, in this presentation, we derive the following (presumably new) limit formulas: lim z ->-k Gamma(nz)/Gamma(qz) = (-1)((n-q)k) (.)q/n . (qk)!/(nk)! and lim z ->-k Psi(nz)/Psi(qz) = q/n for positive integer values of n and q, k being a non-negative integer. One of the above limit formulas is used here to partition the singularities of Gamma(z) into the first set of singularities (at z = 0, -1, -2, -3) and the second set of singularities (at z = -4, -5, -6,...). Some other closely-related formulas are also considered.
引用
收藏
页码:587 / 592
页数:6
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