scholarly journals Accurate analytic approximation for the Chapman grazing incidence function

2021 ◽  
Vol 73 (1) ◽  
Author(s):  
Dmytro Vasylyev
Keyword(s):  
Asymptotic Expansion ◽  
Radiative Transfer ◽  
Bessel Functions ◽  
Grazing Incidence ◽  
Analytical Approximation ◽  
Analytic Approximation ◽  
Atmospheric Physics ◽  
Uniform Asymptotic Expansion ◽  
Grazing Angles ◽  
Physical And Chemical

AbstractA new analytical approximation for the Chapman mapping integral, $${\text {Ch}}$$ Ch , for exponential atmospheres is proposed. This formulation is based on the derived relation of the Chapman function to several classes of the incomplete Bessel functions. Application of the uniform asymptotic expansion to the incomplete Bessel functions allowed us to establish the precise analytical approximation to $${\text {Ch}}$$ Ch , which outperforms established analytical results. In this way the resource consuming numerical integration can be replaced by the derived approximation with higher accuracy. The obtained results are useful for various branches of atmospheric physics such as the calculations of optical depths in exponential atmospheres at large grazing angles, physical and chemical aeronomy, atmospheric optics, ionospheric modeling, and radiative transfer theory.

Global asymptotics of the Szegő–Askey polynomials

2014 ◽  
Vol 12 (06) ◽  
pp. 727-746 ◽  
Author(s):  
Y. Lin ◽  
R. Wong
Keyword(s):  
Asymptotic Behavior ◽  
Asymptotic Expansion ◽  
Weight Function ◽  
Unit Circle ◽  
Bessel Functions ◽  
Asymptotic Formulas ◽  
Elementary Functions ◽  
Integral Methods ◽  
Uniform Asymptotic Expansion ◽  
Global Asymptotics

The Szegő–Askey polynomials are orthogonal polynomials on the unit circle. In this paper, we study their asymptotic behavior by knowing only their weight function. Using the Riemann–Hilbert method, we obtain global asymptotic formulas in terms of Bessel functions and elementary functions for z in two overlapping regions, which together cover the whole complex plane. Our results agree with those obtained earlier by Temme [Uniform asymptotic expansion for a class of polynomials biorthogonal on the unit circle, Constr. Approx. 2 (1986) 369–376]. Temme's approach started from an explicit expression of the Szegő–Askey polynomials in terms of an2F1-function, and followed by integral methods.

Integrals with nearly coincident branch points: Gegenbauer polynomials of large degree

2006 ◽  
Vol 463 (2079) ◽  
pp. 697-710 ◽  
Author(s):  
F Ursell
Keyword(s):  
Asymptotic Expansion ◽  
Bessel Functions ◽  
Complex Function ◽  
Maximum Modulus ◽  
Large Degree ◽  
Branch Points ◽  
Complex Function Theory ◽  
Complete Asymptotic Expansion ◽  
Uniform Asymptotic Expansion ◽  
Standard Techniques

The integral considered here is a loop integral of the form where the integrand has branch points at t =± θ , F ( t 2 ,  θ 2 ) is an analytic function of its arguments and N is a large positive parameter. When θ is not small, its complete asymptotic expansion can be found by standard techniques. When θ is small, the branch points are nearly coincident, and it will be shown that there is a uniform asymptotic expansion involving Bessel functions of argument Nθ . An inequality will be established and will be used to show that the expansion is valid in a region including a disc | θ |≤ m of the complex θ -plane, where m does not tend to 0, when N tends to ∞. The proof of this inequality uses the maximum-modulus principle of complex function theory.

Asymptotic expansions for the coefficient functions that arise in turning-point problems

1987 ◽  
Vol 410 (1838) ◽  
pp. 35-60 ◽  
Keyword(s):  
Differential Equation ◽  
Asymptotic Expansion ◽  
Turning Point ◽  
Bessel Functions ◽  
Holomorphic Functions ◽  
Standard Theory ◽  
Large Parameter ◽  
Uniform Asymptotic Expansion ◽  
All Solutions ◽  
Linear Differential

We study the uniform asymptotic expansion for a large parameter u of solutions of second-order linear differential equations in domains of the complex plane which include a turning point. The standard theory of Olver demonstrates the existence of three such expansions, corresponding to solutions of the form Ai ( u 2/3 ζ e 2/3απi ) Ʃ n s = 0 A s (ζ)/ u 2 s + u -2 d / d ζ Ai ( u 2/3 ζ e 2/3απi ) Ʃ n -1 s =0 B s (ζ)/u 2 s + ϵ ( α ) n ( u , ζ) for α = 0, 1, 2, with bounds on ϵ ( α ) n . We proceed differently, by showing that the set of all solutions of the differential equation is of the form Ai ( u 2/3 ζ) A ( u , ζ) + u -2 (d/dζ) Ai ( u 2/3 ζ) B ( u , ζ), where Ai denotes any situation of Airy's equation. The coefficent functions A ( u , ζ) and B ( u , ζ) are the focus of our attention : we show that for sufficiently large u they are holomorphic functions of ζ in a domain including the turning point, and as functions of u are described asymptotically by descending power series in u 2 , with explicit error bounds. We apply our theory to Bessel functions.

ESTIMATES FOR THE ERROR TERM IN A UNIFORM ASYMPTOTIC EXPANSION OF THE JACOBI POLYNOMIALS

2003 ◽  
Vol 01 (02) ◽  
pp. 213-241 ◽  
Author(s):  
R. WONG ◽  
Y.-Q. ZHAO
Keyword(s):  
Asymptotic Expansion ◽  
Explicit Expression ◽  
Canonical Form ◽  
Error Term ◽  
Jacobi Polynomials ◽  
Recursive Formula ◽  
Contour Integral ◽  
Integration By Parts ◽  
Uniform Asymptotic Expansion ◽  
Repeated Use

There are now several ways to derive an asymptotic expansion for [Formula: see text], as n → ∞, which holds uniformly for [Formula: see text]. One of these starts with a contour integral, involves a transformation which takes this integral into a canonical form, and makes repeated use of an integration-by-parts technique. There are two advantages to this approach: (i) it provides a recursive formula for calculating the coefficients in the expansion, and (ii) it leads to an explicit expression for the error term. In this paper, we point out that the estimate for the error term given previously is not sufficient for the expansion to be regarded as genuinely uniform for θ near the origin, when one takes into account the behavior of the coefficients near θ = 0. Our purpose here is to use an alternative method to estimate the remainder. First, we show that the coefficients in the expansion are bounded for [Formula: see text]. Next, we give an estimate for the error term which is of the same order as the first neglected term.

Exponential sums twisted by Fourier coefficients of automorphic cusp forms for SL(2, ℤ)

2014 ◽  
Vol 11 (01) ◽  
pp. 39-49 ◽  
Author(s):  
Bin Wei
Keyword(s):  
Asymptotic Expansion ◽  
Cusp Form ◽  
Rational Number ◽  
Fourier Coefficients ◽  
Bessel Functions ◽  
Exponential Sums ◽  
Summation Formula ◽  
Cusp Forms ◽  
Real Numbers ◽  
Voronoi Summation

Let f be a holomorphic cusp form of weight k for SL(2, ℤ) with Fourier coefficients λf(n). We study the sum ∑n>0λf(n)ϕ(n/X)e(αn), where [Formula: see text]. It is proved that the sum is rapidly decaying for α close to a rational number a/q where q2 < X1-ε. The main techniques used in this paper include Dirichlet's rational approximation of real numbers, a Voronoi summation formula for SL(2, ℤ) and the asymptotic expansion for Bessel functions.

On uniform asymptotic expansions of finite Laplace and Fourier integrals

1980 ◽  
Vol 85 (3-4) ◽  
pp. 299-305 ◽  
Author(s):  
Kusum Soni
Keyword(s):  
Asymptotic Expansion ◽  
Asymptotic Expansions ◽  
Fourier Integral ◽  
Uniform Asymptotic Expansion ◽  
Laplace Integrals ◽  
Uniform Asymptotic Expansions ◽  
Fourier Integrals

SynopsisA uniform asymptotic expansion of the Laplace integrals ℒ(f, s) with explicit remainder terms is given. This expansion is valid in the whole complex s−plane. In particular, for s = −ix, it provides the Fourier integral expansion.

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