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.

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 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.

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.

scholarly journals On the asymptotic expansion of the logarithm of Barnes triple gamma function

2009 ◽  
Vol 105 (2) ◽  
pp. 287 ◽  
Author(s):  
Stamatis Koumandos ◽  
Henrik L. Pedersen
Keyword(s):  
Asymptotic Expansion ◽  
Gamma Function ◽  
Error Bounds ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions

It is shown that the remainders in an asymptotic expansion of the logarithm of Barnes triple gamma function give rise to completely monotonic functions. Furthermore, error bounds are found.

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 Oscillating behaviour of the spectrum for a plasmonic problem in a domain with a rounded corner

2018 ◽  
Vol 52 (4) ◽  
pp. 1285-1313 ◽  
Author(s):  
Lucas Chesnel ◽  
Xavier Claeys ◽  
Sergei A. Nazarov
Keyword(s):  
Asymptotic Expansion ◽  
Eigenvalue Problem ◽  
Recent Work ◽  
Present Article ◽  
Error Estimates ◽  
Numerical Experiments ◽  
Source Term ◽  
Oscillatory Behaviour ◽  
Time Harmonic ◽  
Rounded Corner

We investigate the eigenvalue problem −div(σ∇u) = λu (P) in a 2D domain Ω divided into two regions Ω±. We are interested in situations where σ takes positive values on Ω+ and negative ones on Ω−. Such problems appear in time harmonic electromagnetics in the modeling of plasmonic technologies. In a recent work [L. Chesnel, X. Claeys and S.A. Nazarov, Asymp. Anal. 88 (2014) 43–74], we highlighted an unusual instability phenomenon for the source term problem associated with (P): for certain configurations, when the interface between the subdomains Ω± presents a rounded corner, the solution may depend critically on the value of the rounding parameter. In the present article, we explain this property studying the eigenvalue problem (P). We provide an asymptotic expansion of the eigenvalues and prove error estimates. We establish an oscillatory behaviour of the eigenvalues as the rounding parameter of the corner tends to zero. We end the paper illustrating this phenomenon with numerical experiments.

On the asymptotic expansion of the logarithm of Barnes triple gamma function II

2017 ◽  
Vol 120 (2) ◽  
pp. 291
Author(s):  
Stamatis Koumandos ◽  
Henrik L. Pedersen
Keyword(s):  
Asymptotic Expansion ◽  
Gamma Function ◽  
Positive Order ◽  
Monotonic Functions ◽  
Completely Monotonic ◽  
Completely Monotonic Functions

The remainders in an asymptotic expansion of the logarithm of Barnes triple gamma function give rise to completely monotonic functions of positive order.

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