Corpus ID: 117202002

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}
}
  • V. Kowalenko
  • Published 2014
  • Mathematics
  • arXiv: Classical Analysis and ODEs
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

Exact Values of the Gamma Function from Stirling’s Formula
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 gammaExpand
Comments on "Exactification of Stirling's approximation for the logarithm of the gamma function"
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’sExpand
Analyzing and provably improving fixed budget ranking and selection algorithms
TLDR
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"
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’sExpand
Exactification of the asymptotics for Bessel and Hankel functions
  • V. Kowalenko
  • Mathematics, Computer Science
  • Appl. Math. Comput.
  • 2002
TLDR
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
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 typicallyExpand
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
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 StirlingExpand
Hyperasymptotics for integrals with saddles
  • M. Berry, C. Howls
  • 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 exponentiallyExpand
Uniform asymptotic smoothing of Stokes’s discontinuities
  • M. Berry
  • 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 itsExpand
Properties and Applications of the Reciprocal Logarithm Numbers
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 coefficientsExpand
Euler and Divergent Series
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 shouldExpand
Generalizing the Reciprocal Logarithm Numbers by Adapting the Partition Method for a Power Series Expansion
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 thisExpand
Exponentially-improved asymptotics for the gamma function
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
...
1
2
3
4
...