The asymptotic expansion of the Bateman and Havelock functions of large order and argument

2021/09/01 by R. B. Paris, Paris, R B
Mathematics · #30E20 #33E20 #34E05 #41A60 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Fractional Differential Equations Solutions #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials

paper · pdf · doi:10.48550/arxiv.2109.00529

openalex publication_date 2021/09/01 · openalex created_date 2021/09/13 · openalex updated_date 2026/07/28

Abstract

Asymptotic expansions for the Bateman and Havelock functions defined respectively by the integrals \frac2π∫₀^π/2 cos
sin (xtan u-νu) du are obtained for large real x and large order ν>0 when ν=O(|x|). The expansions are obtained by application of the method of steepest descents combined with an inversion process to determine the coefficients. Numerical results are presented to illustrate the accuracy of the different expansions obtained.

Related

• Asymptotics of some generalised sine-integrals
• Large-degree asymptotic expansions for the Jacobi and allied functions
• Uniform asymptotic expansions for Bessel functions of imaginary order and their zeros
• Uniform asymptotic expansions of the Jacobi polynomials and an associated function
• Uniform asymptotic expansions for the zeros of Bessel functions