On the large argument asymptotics of the Lommel function via Stieltjes transforms

Abstract

The aim of this paper is to investigate in detail the known large argument asymptotic series of the Lommel function by Stieltjes transform representations. We obtain a number of properties of this asymptotic expansion, including explicit and realistic error bounds, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities. An interesting consequence related to the large argument asymptotic series of the Struve function is also proved.2010 Mathematics Subject Classification. 41A60, 30E15, 34M40.

Error Bounds for the Large‐Argument Asymptotic Expansions of the Lommel and Allied Functions

“…Throughout this paper, if not stated otherwise, empty sums are taken to be zero. The derivations of the estimates for R The remainder R (S) N (z, μ, ν) has already been investigated in an earlier paper of the author [3]. The results of the paper [3] are special cases or consequences of the results of the present work.…”

“…Here, 0 < θ (S) N (z, μ, ν) < 1 is an appropriate number that depends on z, μ, ν, and N (see [3]). Said differently, the remainder term R (S) N (z, μ, ν) does not exceed the corresponding first neglected term in absolute value and has the same sign, provided that z > 0 and μ + |ν| < 2N + 1.…”

“…Indeed, the index of the numerically least term of the asymptotic expansion (1), for example, is n ≈ 1 2 |z|. Therefore, it is reasonable to choose the optimal N ≈ 1 2 |z|, whereas the condition μ ± ν = o(|z|) has to be fulfilled in order to obtain proper approximations from (3).…”

Uniform asymptotic expansions for Lommel, Anger‐Weber, and Struve functions

“…Our new functions do not satisfy the reflection formula (11), but instead, from that relation along with (10), (11)…”

Uniform asymptotic expansions for Lommel, Anger-Weber and Struve functions

“…In [14] he obtained asymptotic expansions of a generalised Struve function for large complex argument. In [9] and [10] Nemes studied the the Lommel and related functions in detail for large z and bounded parameters, deriving explicit error bounds, exponentially improved asymptotic expansions, and the smooth transition of the Stokes discontinuities.…”