scholarly journals Incomplete Laplace Integrals: Uniform Asymptotic Expansion with Application to the Incomplete Beta Function

1987 ◽  
Vol 18 (6) ◽  
pp. 1638-1663 ◽  
Author(s):  
N. M. Temme
Keyword(s):  
Asymptotic Expansion ◽  
Beta Function ◽  
Incomplete Beta Function ◽  
Uniform Asymptotic Expansion ◽  
Laplace Integrals

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.

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.

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