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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The principal inverse of the gamma function


Author: Mitsuru Uchiyama
Journal: Proc. Amer. Math. Soc. 140 (2012), 1343-1348
MSC (2010): Primary 33B15; Secondary 26A48, 47A63
Published electronically: August 3, 2011
MathSciNet review: 2869117
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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)\vert _{(\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 $ {\bf 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 $ {\bf 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
Email: uchiyama@riko.shimane-u.ac.jp

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-11023-2
PII: S 0002-9939(2011)11023-2
Keywords: Gamma function, Pick function, Nevanlinna function, Loewner kernel function
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.