Some asymptotics for the Bessel functions with an explicit error term

2011/07/11 by Ilia Krasikov, Krasikov, Ilia · 1 citation
Mathematics · #Applied mathematics #Bessel function #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Mathematical analysis #Mathematical functions and polynomials #Mathematics #Orthogonal polynomials #Physics #Quantum mechanics #Spectral Theory in Mathematical Physics #Struve function #Term (time) #math.CA

paper · pdf · doi:10.48550/arxiv.1107.2007

Typos corrected

openalex publication_date 2011/07/11 · arxiv created 2011/07/14 · arxiv updated 2011/07/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show how one can obtain an asymptotic expression for some special functions satisfying a second order differential equation with a very explicit error term starting from appropriate upper bounds. We will work out the details for the Bessel function J_ν(x) and the Airy function Ai(x) and find a sharp approximation for their zeros. We also answer the question raised by Olenko by showing that c₁ | ν²-1/4 | < sup_{x ≥ 0} x^3/2|J_ν(x)-√((2)/(πx)) cos (x-(πν)/(2)-\fracπ4 )|

Citations

• Classical and Quantum Orthogonal Polynomials in One Variable

Cited by

• Spectral density estimates of surface-localized eigenmodes for transmission eigenvalue problems

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