scholarly journals Error bounds and exponential improvements for the asymptotic expansions of the gamma function and its reciprocal

2015 ◽  
Vol 145 (3) ◽  
pp. 571-596 ◽  
Author(s):  
Gergő Nemes
Keyword(s):  
Asymptotic Expansion ◽  
Error Estimates ◽  
Gamma Function ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Asymptotic Series ◽  
Smooth Transition ◽  
Error Terms

In 1994 Boyd derived a resurgence representation for the gamma function, exploiting the 1991 reformulation of the method of steepest descents by Berry and Howls. Using this representation, he was able to derive a number of properties of the asymptotic expansion for the gamma function, including explicit and realistic error bounds, the smooth transition of the Stokes discontinuities and asymptotics for the late coefficients. The main aim of this paper is to modify Boyd’s resurgence formula, making it suitable for deriving better error estimates for the asymptotic expansions of the gamma function and its reciprocal. We also prove the exponentially improved versions of these expansions complete with error terms. Finally, we provide new (formal) asymptotic expansions for the coefficients appearing in the asymptotic series and compare their numerical efficacy with the results of earlier authors.

scholarly journals Error bounds and exponential improvement for Hermite's asymptotic expansion for the Gamma function

2013 ◽  
Vol 7 (1) ◽  
pp. 161-179 ◽  
Author(s):  
Gergő Nemes
Keyword(s):  
Asymptotic Expansion ◽  
Error Estimates ◽  
Gamma Function ◽  
Error Bounds ◽  
Asymptotic Series

In this paper we reconsider the asymptotic expansion of the Gamma function with shifted argument, which is the generalization of the well-known Stirling series. To our knowledge, no explicit error bounds exist in the literature for this expansion. Therefore, the first aim of this paper is to extend the known error estimates of Stirling?s series to this general case. The second aim is to give exponentially-improved asymptotics for this asymptotic series.

scholarly journals Error bounds for the asymptotic expansion of the Hurwitz zeta function

2017 ◽  
Vol 473 (2203) ◽  
pp. 20170363 ◽  
Author(s):  
G. Nemes
Keyword(s):  
Asymptotic Expansion ◽  
Zeta Function ◽  
Gamma Function ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Remainder Term ◽  
Hurwitz Zeta Function ◽  
Polygamma Functions

In this paper, we reconsider the large- a asymptotic expansion of the Hurwitz zeta function ζ ( s , a ). New representations for the remainder term of the asymptotic expansion are found and used to obtain sharp and realistic error bounds. Applications to the asymptotic expansions of the polygamma functions, the gamma function, the Barnes G -function and the s -derivative of the Hurwitz zeta function ζ ( s , a ) are provided. A detailed discussion on the sharpness of our error bounds is also given.

scholarly journals The resurgence properties of the incomplete gamma function, I

2016 ◽  
Vol 14 (05) ◽  
pp. 631-677 ◽  
Author(s):  
Gergő Nemes
Keyword(s):  
Gamma Function ◽  
Error Bounds ◽  
Asymptotic Expansions ◽  
Incomplete Gamma Function ◽  
Smooth Transition

In this paper, we derive new representations for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by C. J. Howls [Hyperasymptotics for integrals with finite endpoints, Proc. Roy. Soc. London Ser. A 439 (1992) 373–396]. Using these representations, we obtain a number of properties of the asymptotic expansions of the incomplete gamma function with large arguments, including explicit and realistic error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.

scholarly journals Prime geodesics and averages of the Zagier L-series

2021 ◽  
pp. 1-24
Author(s):  
OLGA BALKANOVA ◽  
DMITRY FROLENKOV ◽  
MORTEN S. RISAGER
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Error Term ◽  
Quadratic Fields ◽  
Real Quadratic Fields ◽  
Prime Geodesic Theorem ◽  
Error Terms ◽  
Average Size ◽  
Real Quadratic

Abstract The Zagier L-series encode data of real quadratic fields. We study the average size of these L-series, and prove asymptotic expansions and omega results for the expansion. We then show how the error term in the asymptotic expansion can be used to obtain error terms in the prime geodesic theorem.

Corrigendum and addendum to ‘asymptotic series for double zeta, double gamma and Hecke L-functions’

2002 ◽  
Vol 132 (2) ◽  
pp. 377-384 ◽  
Author(s):  
KOHJI MATSUMOTO
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Series ◽  
Gamma Functions ◽  
Double Zeta ◽  
Error Terms

Refined expressions are given for the error terms in the asymptotic expansion formulas for double zeta and double gamma functions, proved in the author's former paper [2]. Some inaccurate claims in [2] are corrected.

scholarly journals The resurgence properties of the large-order asymptotics of the Hankel and Bessel functions

2014 ◽  
Vol 12 (04) ◽  
pp. 403-462 ◽  
Author(s):  
Gergő Nemes
Keyword(s):  
Error Bounds ◽  
Asymptotic Expansions ◽  
Bessel Functions ◽  
Smooth Transition ◽  
Large Order ◽  
Computable Error Bounds

The aim of this paper is to derive new representations for the Hankel and Bessel functions, exploiting the reformulation of the method of steepest descents by Berry and Howls [Hyperasymptotics for integrals with saddles, Proc. R. Soc. Lond. A 434 (1991) 657–675]. Using these representations, we obtain a number of properties of the large-order asymptotic expansions of the Hankel and Bessel functions due to Debye, including explicit and numerically computable error bounds, asymptotics for the late coefficients, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.

scholarly journals Exactification of the Poincaré asymptotic expansion of the Hankel integral: spectacularly accurate asymptotic expansions and non-asymptotic scales

2014 ◽  
Vol 470 (2162) ◽  
pp. 20130529 ◽  
Author(s):  
Eric A. Galapon ◽  
Kay Marie L. Martinez
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Function ◽  
Asymptotic Expansions ◽  
Integral Method ◽  
Asymptotic Series ◽  
Poincaré Series ◽  
Poincare Series ◽  
Half Integer ◽  
Distributional Method ◽  
Finite Sum

We obtain an exactification of the Poincaré asymptotic expansion (PAE) of the Hankel integral, as , using the distributional approach of McClure & Wong. We find that, for half-integer orders of the Bessel function, the exactified asymptotic series terminates, so that it gives an exact finite sum representation of the Hankel integral. For other orders, the asymptotic series does not terminate and is generally divergent, but is amenable to superasymptotic summation, i.e. by optimal truncation. For specific examples, we compare the accuracy of the optimally truncated asymptotic series owing to the McClure–Wong distributional method with owing to the Mellin–Barnes integral method. We find that the former is spectacularly more accurate than the latter, by, in some cases, more than 70 orders of magnitude for the same moderate value of b . Moreover, the exactification can lead to a resummation of the PAE when it is exact, with the resummed Poincaré series exhibiting again the same spectacular accuracy. More importantly, the distributional method may yield meaningful resummations that involve scales that are not asymptotic sequences.

Gamma function asymptotics by an extension of the method of steepest descents

1994 ◽  
Vol 447 (1931) ◽  
pp. 609-630 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Recent Work ◽  
Gamma Function ◽  
Error Bounds ◽  
Stokes Line ◽  
Asymptotic Estimates ◽  
Asymptotic Representations

Recent work of Berry & Howls, which reformulated the method of steepest de­scents, is exploited to derive a new representation for the gamma function. It is shown how this representation can be used to derive a number of properties of the asymptotic expansion of the gamma function, including explicit and realistic error bounds, the Berry transition between different asymptotic representations across a Stokes line, and asymptotic estimates for the late coefficients.

Infinitely many Stokes smoothings in the gamma function

1991 ◽  
Vol 434 (1891) ◽  
pp. 465-472 ◽  
Keyword(s):  
Asymptotic Expansion ◽  
Gamma Function ◽  
Error Function ◽  
Asymptotic Series ◽  
Stokes Line ◽  
Separate Component ◽  
Negative Real Axis ◽  
Stokes Lines ◽  
The Right ◽  
Anti Stokes

The Stokes lines for Г( z ) are the positive and negative imaginary axes, where all terms in the divergent asymptotic expansion for In Г( z ) have the same phase. On crossing these lines from the right to the left half-plane, infinitely many subdominant exponentials appear, rather than the usual one. The exponentials increase in magnitude towards the negative real axis (anti-Stokes line), where they add to produce the poles of Г( z ). Corresponding to each small exponential is a separate component asymptotic series in the expansion for In Г( z ). If each is truncated near its least term, its exponential switches on smoothly across the Stokes lines according to the universal error-function law. By appropriate subtractions from In Г( z ), the switching-on of successively smaller exponentials can be revealed. The procedure is illustrated by numerical computations.

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