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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The principal inverse of the gamma function
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by Mitsuru Uchiyama PDF
Proc. Amer. Math. Soc. 140 (2012), 1343-1348 Request permission

Abstract:

Let $\Gamma (x)$ be the gamma function in the real axis and $\alpha$ the maximal zero of $\Gamma ’(x)$. We call the inverse function of $\Gamma (x)|_{(\alpha , \infty )}$ the principal inverse and denote it by $\Gamma ^{-1}(x)$. We show that $\Gamma ^{-1}(x)$ has the holomorphic extension $\Gamma ^{-1}(z)$ to $\textbf {C}\setminus (-\infty , \Gamma (\alpha )]$, which maps the upper half-plane into itself, namely a Pick function, and that $\Gamma (\Gamma ^{-1}(z))= z$ on $\textbf {C}\setminus (-\infty , \Gamma (\alpha )].$
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Additional Information
  • Mitsuru Uchiyama
  • Affiliation: Department of Mathematics, Interdisciplinary Faculty of Science and Engineering, Shimane University, Matsue City, Shimane 690-8504, Japan
  • MR Author ID: 198919
  • Email: uchiyama@riko.shimane-u.ac.jp
  • Received by editor(s): November 18, 2010
  • Received by editor(s) in revised form: January 4, 2011
  • Published electronically: August 3, 2011
  • Additional Notes: The author was supported in part by (JSPS) KAKENHI 21540181
  • Communicated by: Richard Rochberg
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 1343-1348
  • MSC (2010): Primary 33B15; Secondary 26A48, 47A63
  • DOI: https://doi.org/10.1090/S0002-9939-2011-11023-2
  • MathSciNet review: 2869117