# Gamma and Factorial in the Monthly

@article{Borwein2018GammaAF, title={Gamma and Factorial in the Monthly}, author={Jonathan Michael Borwein and Robert M Corless}, journal={The American Mathematical Monthly}, year={2018}, volume={125}, 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…

## 15 Citations

Stirling’s Original Asymptotic Series from a Formula Like One of Binet’s and its Evaluation by Sequence Acceleration

- MathematicsExperimental Mathematics
- 2019

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

- Mathematics
- 2018

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

- Computer Science, MathematicsBIT Numerical Mathematics
- 2019

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

- Computer ScienceArXiv
- 2021

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

- MathematicsAm. Math. Mon.
- 2019

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

- Mathematics
- 2021

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

- Mathematics
- 2019

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($F_n$)

- MathematicsCommentarii Mathematici Helvetici
- 2020

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 $\operatorname{Out}(F_n)$

- Mathematics
- 2019

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.

- Mathematics
- 2020

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…

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