Gamma and Factorial in the Monthly

Abstract

Since its inception in 1894, the Monthly has printed 50 articles on the Γ function or Stirling's asymptotic formula, including the magisterial 1959 paper by Phillip J. Davis, which won the 1963 Chauvenet prize, and the eye-opening 2000 paper by the Fields medalist Manjul Bhargava. In this article, we look back and comment on what has been said, and why, and try to guess what will be said about the Γ function in future Monthly issues.¹

1 It is a safe bet that there will be more proofs of Stirling's formula, for instance.

We also identify some gaps, which surprised us: phase plots, Riemann surfaces, and the functional inverse of Γ make their first appearance in the Monthly here. We also give a new elementary treatment of the asymptotics of n! and the first few terms of a new asymptotic formula for invΓ.

MSC:

• Primary 01-02
• Secondary 33-02; 33-03; 33B15

Acknowledgments

This work was supported by Western University through a Distinguished Visiting Research Fellowship for the first author, and by NSERC, the Fields Institute for Research in the Mathematical Sciences, the Rotman Institute for Philosophy, and the Ontario Research Centre for Computer Algebra for the second. The manuscript was typed by Julia E. Jankowski and Chris Brimacombe. We also thank the referees for many constructive comments, and the editors for painstaking corrections.

Notes

1 It is a safe bet that there will be more proofs of Stirling's formula, for instance.

2 In [13] the authors highlighted what they called “π*” articles, i.e., those that had been cited at least 30 times. In contrast, inclusion in this list here is purely subjective. This list does correlate to citation, though.

3 As defined in the introduction, pseudogamma functions satisfy f(x + 1) = xf(x). Hadamard's function satisfies H(x + 1) = xH(x) + 1/Γ(1 − x), so it interpolates Γ only at the positive integers.

4 It is interesting that the notion “an average paper in the Monthly on Γ” actually makes sense.

5 Like the determinant, but with all plus signs instead of ( − 1)^σ in the permutation expansion.

6 The details of the code are available at apmaths.uwo.ca/∼rcorless/frames/Gamma/GammaBorweinCorless.mw

7 Which could be applied to z! and log Γ(1 + z). See [26].

8 Some proofs do appear in the Monthly, e.g., Blyth and Pathak [12] use Γ(n + α) ∼ n^αΓ(n).

9 We found [46] by browsing the Monthly, not by citation search. Its criticism of [66] is largely sound: Copson's treatment is better in many respects.

10 This is in several texts, including [22]—but one place to learn it is by teaching from [97].

11 One of the referees said that equation (53(53) ) was “obviously mistyped” and “does not make sense.” We believe that it is as Stirling wrote it, but the notation ℓ, x for log ₁₀(x) does look weird to modern eyes, it is true! Perhaps Stirling used the mnemonic “comma ℓ for common log ”?

12 We cannot use an exclamation mark, for obvious reasons.

13 isc.carma.newcastle.edu.au Indeed, Stirling himself may have used “experimental mathematics” by hand to identify the constant , according to [101].

14 We remind the reader that divergent series are often remarkably accurate. There are two relevant limits, only one of which might be bad. However, Stirling's formula is famously accurate, more so than most. And Stirling's original formula is even better, because the midpoint rule is twice as accurate as the trapezoidal rule.

Additional information

Funding

Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada(10.13039/501100002790)

Notes on contributors

Jonathan M. Borwein

JONATHAN M. BORWEIN was Laureate Professor of mathematics at the University of Newcastle, Australia. An ISI highly cited scientist and former Chauvenet Prize winner, he has published widely in various fields of mathematics. For more details, please see his tribute obituary in the November 2016 issue of the Monthly.

Robert M. Corless

ROBERT M. CORLESS is the Scientific Director of the Ontario Research Centre for Computer Algebra (www.orcca.on.ca). He did his B.Sc. in Mathematics and Computer Science at the University of British Columbia, M. Math at Waterloo in Applied Mathematics, and his Ph.D. (1987, in Mechanical Engineering) again at UBC. He has been a professor at the University of Western Ontario ever since. He has written three books and over 175 papers.