# Exactification of Stirling's Approximation for the Logarithm of the Gamma Function

@article{Kowalenko2014ExactificationOS, title={Exactification of Stirling's Approximation for the Logarithm of the Gamma Function}, author={Victor Kowalenko}, journal={arXiv: Classical Analysis and ODEs}, year={2014} }

Exactification is the process of obtaining exact values of a function from its complete asymptotic expansion. Here Stirling's approximation for the logarithm of the gamma function or $\ln \Gamma(z)$ is derived completely whereby it is composed of the standard leading terms and an asymptotic series that is generally truncated. Nevertheless, to obtain values of $\ln \Gamma(z)$, the remainder must undergo regularization. Two regularization techniques are then applied: Borel summation and Mellin… Expand

#### Figures and Tables from this paper

#### 3 Citations

Exact Values of the Gamma Function from Stirling’s Formula

- Mathematics
- 2020

In this work the complete version of Stirling’s formula, which is composed of the standard terms and an infinite asymptotic series, is used to obtain exact values of the logarithm of the gamma… Expand

Comments on "Exactification of Stirling's approximation for the logarithm of the gamma function"

- Mathematics
- 2014

We re-examine the exponentially improved expansion for log ( z), first considered in Paris and Wood in 1991, to point out that the recent treatment by Kowalenko [Exactification of Stirling’s… Expand

Analyzing and provably improving fixed budget ranking and selection algorithms

- Mathematics, Computer Science
- ArXiv
- 2018

This paper focuses on a more tractable two-design case and explicitly characterize the large deviations rate of PFS for some simplified algorithms, and highlights several useful techniques for analyzing the convergence rate of fixed budget R&S algorithms. Expand

#### References

SHOWING 1-10 OF 32 REFERENCES

Comments on "Exactification of Stirling's approximation for the logarithm of the gamma function"

- Mathematics
- 2014

We re-examine the exponentially improved expansion for log ( z), first considered in Paris and Wood in 1991, to point out that the recent treatment by Kowalenko [Exactification of Stirling’s… Expand

Exactification of the asymptotics for Bessel and Hankel functions

- Mathematics, Computer Science
- Appl. Math. Comput.
- 2002

Both techniques for evaluating divergent series with great precision far more rapidly than Borel summation are presented in the evaluation of exact values for Bessel and Hankel functions from their complete asymptotic expansions. Expand

Asymptotics and Mellin-Barnes Integrals

- Mathematics
- 2001

Asymptotics and Mellin-Barnes Integrals provides an account of the use and properties of a type of complex integral representation that arises frequently in the study of special functions typically… Expand

Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π−1

- Mathematics
- 2014

Abstract In this paper, two new series for the logarithm of the Γ-function are presented and studied. Their polygamma analogs are also obtained and discussed. These series involve the Stirling… Expand

Hyperasymptotics for integrals with saddles

- Mathematics
- Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences
- 1991

Integrals involving exp { –kf(z)}, where |k| is a large parameter and the contour passes through a saddle of f(z), are approximated by refining the method of steepest descent to include exponentially… Expand

Uniform asymptotic smoothing of Stokes’s discontinuities

- Mathematics
- Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences
- 1989

Across a Stokes line, where one exponential in an asymptotic expansion maximally dominates another, the multiplier of the small exponential changes rapidly. If the expansion is truncated near its… Expand

Properties and Applications of the Reciprocal Logarithm Numbers

- Mathematics
- 2010

Via a graphical method, which codes tree diagrams composed of partitions, a novel power series expansion is derived for the reciprocal of the logarithmic function ln (1+z), whose coefficients… Expand

Euler and Divergent Series

- Mathematics
- 2011

Euler’s reputation is tarnished because of his views on divergent series. He believed that all series should have a value, not necessarily a limit as for convergent series, and that the value should… Expand

Generalizing the Reciprocal Logarithm Numbers by Adapting the Partition Method for a Power Series Expansion

- Mathematics
- 2009

Recently, a novel method based on the coding of partitions was used to determine a power series expansion for the reciprocal of the logarithmic function, viz. z/ln (1+z). Here we explain how this… Expand

Exponentially-improved asymptotics for the gamma function

- Mathematics
- 1992

By expressing the error term in truncation of the asymptotic expansion in terms of a Mellin-Barnes integral, we obtain an exponentially-improved asymptotic expansion for Г(z) as ∣z ∣ → ∞ in ∣ arg z ∣… Expand