Gergő Nemes

An Explicit Formula for the Coefficients in Laplace’s Method

Laplace’s method is one of the fundamental techniques in the asymptotic approximation of integrals. The coefficients appearing in the resulting asymptotic expansion arise as the coefficients of a convergent or asymptotic series of a function defined in an implicit form. Due to the tedious computation of these coefficients, most standard textbooks on asymptotic approximations of integrals do not give explicit formulas for them. Nevertheless, we can find some more or less explicit representations for the coefficients in the literature: Perron’s formula gives them in terms of derivatives of an explicit function; Campbell, Fröman and Walles simplified Perron’s method by computing these derivatives using an explicit recurrence relation. The most recent contribution is due to Wojdylo, who rediscovered the Campbell, Fröman and Walles formula and rewrote it in terms of partial ordinary Bell polynomials. In this paper, we provide an alternative representation for the coefficients that contains ordinary potential polynomials. The proof is based on Perron’s formula and a theorem of Comtet. The asymptotic expansions of the gamma function and the incomplete gamma function are given as illustrations.

Generalization of Binet's Gamma function formulas

Several representations for the logarithm of the Gamma function exist in the literature. There are four important expansions which bear the name of Binet. Hermite generalized Binets first formula to the logarithm of the Gamma function with shifted argument. The generalization of Binets second formula is apparently not known; however, it follows easily from another result of Hermite. The aim of this paper is to give possible generalizations of the third and fourth Binet formulas.

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

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.

Error bounds and exponential improvement for the asymptotic expansion of the Barnes G-function

In this paper, we establish new integral representations for the remainder term of the known asymptotic expansion of the logarithm of the Barnes G-function. Using these representations, we obtain explicit and numerically computable error bounds for the asymptotic series, which are much simpler than those obtained earlier by other authors. We find that along the imaginary axis, suddenly infinitely many exponentially small terms appear in the asymptotic expansion of the Barnes G-function. Employing one of our representations for the remainder term, we derive an exponentially improved asymptotic expansion for the logarithm of the Barnes G-function, which shows that the appearance of these exponentially small terms is in fact smooth, thereby proving the Berry transition property of the asymptotic series of the G-function.

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

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.

The Resurgence Properties of the Incomplete Gamma Function II

In this paper, we derive a new representation for the incomplete gamma function, exploiting the reformulation of the method of steepest descents by Howls 1992. Using this representation, we obtain numerically computable bounds for the remainder term of the asymptotic expansion of the incomplete gamma function Γ−a,λa with large a and fixed positive λ, and an asymptotic expansion for its late coefficients. We also give a rigorous proof of Dingles formal result regarding the exponentially improved version of the asymptotic series of Γ−a,λa.

Error bounds for the asymptotic expansion of the Hurwitz zeta function

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.

The resurgence properties of the incomplete gamma function, I

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 [Hyperasymtotics for integrals with finite endpoints, Proc. Roy. Soc. London Ser. A439 (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.

On the large argument asymptotics of the Lommel function via Stieltjes transforms

The aim of this paper is to investigate in detail the known large argument asymptotic series of the Lommel function by Stieltjes transform representations. We obtain a number of properties of this asymptotic expansion, including explicit and realistic error bounds, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities. An interesting consequence related to the large argument asymptotic series of the Struve function is also proved.

Error Bounds for the Large-Argument Asymptotic Expansions of the Lommel and Allied Functions: The Large-argument Asymptotics of the Lommel and Allied Functions

In this paper, we reconsider the large-z asymptotic expansion of the Lommel function Sμ,ν(z) and its derivative. New representations for the remainder terms of the asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give reexpansions for these remainder terms and provide their error estimates. Applications to the asymptotic expansions of the Anger–Weber-type functions, the Scorer functions, the Struve functions and their derivatives are provided. The sharpness of our error bounds is discussed in detail, and numerical examples are given.