Uniform, Exponentially Improved, Asymptotic Expansions for the Confluent Hypergeometric Function and Other Integral Transforms

New asymptotic expansions are derived for the incomplete gamma functions and the incomplete beta function. In each case the expansion contains the complementary error function and an asymptotic series. The expansions are uniformly valid with respect to certain domains of the parameters.

We construct asymptotic expansions for the normalised incomplete gamma function Q ( a , z ) = Γ ( a , z ) / Γ ( a ) Q(a,z)=\Gamma (a,z)/\Gamma (a) that are valid in the transition regions, including the case z ≈ a z\approx a , and have simple polynomial coefficients. For Bessel functions, these types of expansions are well known, but for the normalised incomplete gamma function they were missing from the literature. A detailed historical overview is included. We also derive an asymptotic expansion for the corresponding inverse problem, which has importance in probability theory and mathematical statistics. The coefficients in this expansion are again simple polynomials, and therefore its implementation is straightforward. As a byproduct, we give the first complete asymptotic expansion as a → − ∞ a\to -\infty of the unique negative zero of the regularised incomplete gamma function γ ∗ ( a , x ) \gamma ^*(a,x) .

Asymptotic inversion of the incomplete beta function

Chapter 4 - Special functions

We examine the asymptotic nature as k → ∞ of the coefficientsck(�) appearing in the uniform asymptotic expansion of the incomplete gamma function ( a,z) whereis a variable that depends on the ratio z/a. It is shown that this expansion diverges like the familiar factorial divided by a power dependence multiplied by a function fk(�). For values ofnear the real axis, fk(�) is a slowly varying function, but in the left half-plane, there are two lobes situated symmetrically about the negative realaxis in which fk(�) becomes large. The asymptotic expansion of fk(�) as k → ∞ is found to reveal a resurgence-type structure in which the high-order coefficients are related tothe low-order coeffi- cients. Numerical examples are given to illustrate the growth of the coefficients ck(�).

In this paper, we study the uniform asymptotics of the Meixner‐Pollaczek polynomials with varying parameter as , where A > 0 is a constant. Two asymptotic expansions are obtained, which hold uniformly for z in two overlapping regions which together cover the whole complex plane. One involves parabolic cylinder functions, and the other is in terms of elementary functions only. Our approach is based on the steepest descent method for oscillatory Riemann‐Hilbert problems first introduced by Deift and Zhou [1].

Two sets of closed analytic functions are proposed for the approximate calculus of the complete elliptic integrals of the 1st and 2nd kinds in the normal form due to Legendre, their expressions having a remarkable simplicity and accuracy. The special usefulness of the newly proposed formulas consists in that they allow performing the analytic study of variation of the functions in which they appear, using the derivatives (they being expressed in terms of elementary functions only, without any special function; this would mean replacing one difficulty by another of the same kind). Comparative tables of the approximate values so obtained and the exact ones, reproduced from special functions tables are given (vs. the elliptic integrals modulus k). It is to be noticed that both sets of formulas are given neither by spline nor by regression functions, but by asymptotic expansions, the identity with the exact functions being accomplished for the left domain’s end. As for their simplicity, the formulas in k / k' do not need any mathematical table (are purely algebraic). As for their accuracy, the 2nd set, although more intricate, gives more accurate values than the 1st one and extends itself more closely to the right domain’s end.

In this paper, we present explicit and computable error bounds for the asymptotic expansions of the Hermite polynomials with Plancherel–Rotach scale. Three cases, depending on whether the scaled variable lies in the outer or oscillatory interval, or it is the turning point, are considered separately. We introduce the ‘branch cut’ technique to express the error terms as integrals on the contour taken as the one-sided limit of curves approaching the branch cut. This new technique enables us to derive simple error bounds in terms of elementary functions. We also provide recursive procedures for the computation of the coefficients appearing in the asymptotic expansions.

Asymptotic expansions of certain generalized Bernoulli polynomials are obtained, some in terms of elementary functions and others in terms of gamma functions and their derivatives. The latter results can also be written in terms of elementary functions, by using the known asymptotic expansions of the gamma functions, and the leading terms are obtained in this way. The results obtained give the asymptotic form of the coefficients occurring in all the usual central-difference formulae of the calculus of finite differences.

We denote by and the gamma and the incomplete gamma functions, respectively. In this paper we prove some monotonicity results for the gamma function and extend, to , a lower bound established by Elbert and Laforgia (2000) for the function , with , only for .

در این تحقیق به منظور تعیین بهترین منحنی تولید شیر در یک دوره شیردهی از 260709 رکورد روز آزمون تولید شیر متعلق به 30903 رأس گاو هلشتاین شکم اول استان خراسان استفاده شد. اطلاعات مورد استفاده مربوط به گاو‌های 216 گله بود که طی سالهای 1367 الی 1385 زایش داشتند. توابع مورد استفاده شامل تابع گامای ناقص (وود)، تابع ویلمینک، تابع علی‌شفر و تابع لژاندر (با درجه برازش پنجم) بود. معیارهای مورد نظر برای مقایسه توابع شامل مجموع مربعات خطا، میانگین مجموع مربعات خطا، ضریب تعیین، ضریب تعیین تصحیح شده ، ضریب همبستگی و فاکتور تورم واریانس بود. براساس تمام معیارها به استثناء فاکتور تورم واریانس تابع علی- شفر و براساس فاکتور تورم واریانس تابع لژاندر به نحو مطلوب‌تری با داده‌های تولید شیر مطابقت داشتند.

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.

On the use of Hadamard expansions in hyperasymptotic evaluation of Laplace-type integrals. I: real variable

This article is concerned with the asymptotic behaviour of m(λ), the Titchmarsh-Weyl m-coefficient, for the singular eigenvalue equation y“ + (λ − q(x))y = 0 on [0, ∞), as λ →∞ in a sector in the upper half of the complex plane. It is assumed that the potential function q is integrable near 0. A simplified proof is given of a result of Atkinson [7], who derived the first two terms in the asymptotic expansion of m(λ), and a sharper error bound is obtained. Theproof is then generalised to derive subsequent terms in the asymptotic expansion. It is shown that the Titchmarsh-Weyl m-coefficient admits an asymptotic power series expansion if the potential function satisfies some smoothness condition. A simple method to compute the expansion coefficients is presented. The results for the first few coefficients agree with those given by Harris [9].

Our aim here is to give sufficient conditions on the finite element spaces near a point so that the error in the finite element method for the function and its derivatives at the point have exact asymptotic expansions in terms of the mesh parameter h, valid for h sufficiently small. Such expansions are obtained from the so-called asymptotic expansion inequalities valid in $\mathbf{R}^N$ for $N\geq2$, studies by Schatz in [Math. Comp., 67 (1998), pp. 877–899] and [SIAM J. Numer. Anal., 38 (2000), pp. 1269–1293].