An asymptotic for the K-Bessel function using the saddle-point method

2023/02/20 by Jimmy Tseng, Tseng, Jimmy
Mathematics · #33C10 #41A60 #Classical Analysis and ODEs (math.CA) #Differential Equations and Boundary Problems #FOS: Mathematics #Spectral Theory in Mathematical Physics #advanced mathematical theories

paper · pdf · doi:10.48550/arxiv.2302.09962

openalex publication_date 2023/02/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Using the saddle-point method, we compute an asymptotic, as y → ∞, for the K-Bessel function K_{r + i t}(y) with positive, real argument y and of large complex order r+it where r is bounded and t = y sin θ for a fixed parameter 0≤ θ≤ π/2 or t= y \cosh μ for a fixed parameter μ>0. Our method gives an illustrative proof, using elementary tools, of this known result and explains how these asymptotics come about. As part of our proof, we prove a new result, namely a novel integral representation for K_{r + i t}(y) in the case t= y \cosh μ. This integral representation involves only one saddle point.

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