A Property of the Gamma Function at its Singularities

2010/08/12 by Anirudh Prabhu, Prabhu, Anirudh
Mathematics · #FOS: Mathematics #General Mathematics (math.GM) #math.GM

paper · pdf · doi:10.48550/arxiv.1008.2220

arxiv created 2010/08/12 · arxiv updated 2010/08/16

Abstract

The singularities of the Γ function, a meromorphic function on the complex plane, are known to occur at the nonpositive integers. We show, using Euler and Gauss identities, that for all positive integers n and k, lim_{z→ 0} (Γ(nz))/(Γ(z)) = \frac 1 n; \hspace0.4in lim_{z→ -k} (Γ(nz))/(Γ(z)) = \f(-1)^k Γ(k)n² Γ(nk). The above relations add to the list of the known fundamental Gamma function identities.

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