Gamma and Factorial in the Monthly

  title={Gamma and Factorial in the Monthly},
  author={Jonathan Michael Borwein and Robert M Corless},
  journal={The American Mathematical Monthly},
  pages={400 - 424}
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.11 It is a safe bet that there will be more proofs of… 
Stirling’s Original Asymptotic Series from a Formula Like One of Binet’s and its Evaluation by Sequence Acceleration
Abstract We give an apparently new proof of Stirling’s original asymptotic formula for the behavior of for large z. Stirling’s original formula is not the formula widely known as “Stirling’s
Regularized Integral Representations of the Reciprocal Γ Function
Applications of the Gamma function are ubiquitous in fractional calculus and the special function 7 theory. It has numerous interesting properties summarized in [1]. It is indispensable in the theory
Computation of matrix gamma function
This research article proposes a fourth technique based on the reciprocal gamma function that is shown to be competitive with the other three methods in terms of accuracy, with the advantage of being rich in matrix multiplications.
Arbitrary-precision computation of the gamma function
The best methods available for computing the gamma function Γ(z) in arbitrary-precision arithmetic with rigorous error bounds are discussed and some new formulas, estimates, bounds and algorithmic improvements are presented.
Three Theorems of Menelaus
Three theorems due to Menelaus of Alexandria (1st–2nd century A.D.) that concern spherical triangles are presented and his methods of proof more widely known.
Weighted Prime Powers Truncation of the Asymptotic Expansion for the Logarithmic Integral: Properties and Applications
ABSTRACT. Letting the truncation point for the Logarithmic Integral’s asymptotic expansion be the variable to solve for, though not to be minimized as with the usual Stieltjes truncation. Instead
Asymptotics of some generalized Mathieu series
We establish asymptotic estimates of Mathieu-type series defined by sequences with power-logarithmic or factorial behavior. By taking the Mellin transform, the problem is mapped to the singular
The Euler characteristic of Out($F_n$)
We prove that the rational Euler characteristic of Out(Fn) is always negative and its asymptotic growth rate is Γ(n− 3 2 )/ √ 2π log n. This settles a 1987 conjecture of J. Smillie and the second
The Euler characteristic of $\operatorname{Out}(F_n)$
We prove that the rational Euler characteristic of $\operatorname{Out}(F_n)$ is always negative and its asymptotic growth rate is $\Gamma(n- \frac32)/\sqrt{2\pi} \log^2 n$. This settles a 1987
Notes on the historical bibliography of the gamma function.
Telegraphic notes on the historical bibliography of the Gamma function and Eulerian integrals. Correction to some classical references. Some topics of the interest of the author. We provide some


James Stirling's Methodus differentialis
Contains not only the results and ideas for which Stirling is chiefly remembered, for example, Stirling numbers and Stirling's asymptotic formula for factorials, but also a wealth of material on
A simple proof of Stirling's formula for the gamma function
  • G. Jameson
  • Mathematics
    The Mathematical Gazette
  • 2015
where and the notation means that as . C = 2π f (n) ∼ g(n) f (n) / g(n) → 1 n → ∞ A great deal has been written about Stirling's formula. At this point I will just mention David Fowler's Gazette
Laplace's Integral, the Gamma Function, and beyond
This article focuses on Laplace's basic result and his identity, and avoids popular techniques: "messy" contour integration and the theory of Fourier transforms, which invariably invokes the Fourier inversion formula.
The Gamma Function: An Eclectic Tour
The gamma function was subjected to intense study by almost every eminent mathematician of the nineteenth and early twentieth centuries and continues to interest the present genera tion from the integral representation numerous identities including the beta-gamma relation and the duplication formula of Legendre were deduced.
Note on the Beta and Gamma Functions
In N. Bourbaki, IDlements de Mathematique, Fonctions d'une variable reelle, Ch. 1, 2, 3, 2me Md., 1958, p. 127, one finds a simple and interesting method of evaluating the Euler-Poisson integral f
The (n + 1)th Proof of Stirling's Formula
In the following the author uses a skillful but not so obvious estimate of the In function to present a really elementary deduction of Stirling's formula.
Stirling's Series Made Easy
The method is adapted to obtain Stirling's series for the logarithm of the gamma function and it is shown that the constants obtained are the best ones possible, i.e., they cannot be improved by any method whatsoever.
Graphing elementary Riemann surfaces
This paper shows how to use a computer algebra system (or even a purely numerical graphing package) to graph the Riemann surfaces of various elementary functions.
James Stirling’s Methodus Differentialis: An annotated translation of Stirling’s text, by Ian Tweddle. Pp. 295. €129.95, £ 75.00, sFr 210.00, $ 129.00. 2003. ISBN 1 85233 723 0 (Springer).
classroom as well as Anthony Ferzola's account of Euler's use of differentials as 'absolute zeros', a non-rigorous intuitive approach which was exonerated in retrospect by Abraham Robinson's
Stirling's Formula via the Poisson Distribution
  • M. Pinsky
  • Mathematics, Computer Science
    Am. Math. Mon.
  • 2007
9. A. I. Katsevich and A. G. Ramm, The Radon Transform and Local Tomography, CRC Press, Boca Raton, FL, 1996. 10. E. Lindel?f, Calcul des R?sidus, Gauthier-Villars, Paris, 1905. 11. G.