On the uniform asymptotic expansion of the Legendre functions

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.

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

Analysis and Applications ◽

10.1142/s0219530503000107 ◽

2003 ◽

Vol 01 (02) ◽

pp. 213-241 ◽

Cited By ~ 7

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.

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On uniform asymptotic expansions of finite Laplace and Fourier integrals

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500011872 ◽

1980 ◽

Vol 85 (3-4) ◽

pp. 299-305 ◽

Cited By ~ 4

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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