Showing papers by "Roderick Wong published in 1999"
TL;DR: In this paper, it was shown that for ν > 0 and k = 1, 2, 3,..., ν − ak 21/3 ν < jν,k < ν + απππ απ βππ β β β απ αβ β αβ αββ β ββββ α β α ββ βα ββα β βα αβα α β β π ββ α α βαβ ββδ αβγ ββγ α βγ βγ α
Abstract: Let jν,k denote the k-th positive zero of the Bessel function Jν(x). In this paper, we prove that for ν > 0 and k = 1, 2, 3, . . . , ν − ak 21/3 ν < jν,k < ν − ak 21/3 ν + 3 20 ak 21/3 ν1/3 . These bounds coincide with the first few terms of the well-known asymptotic expansion jν,k ∼ ν − ak 21/3 ν + 3 20 ak 21/3 ν1/3 + · · · as ν →∞, k being fixed, where ak is the k-th negative zero of the Airy function Ai(x), and so are “best possible”.
60 citations
TL;DR: In this article, the authors consider the case -1 < φ < 0 and show that contributions towards hyperasymptotic expansions not only come from adjacent saddles but also from curves that are not even steepest-descent paths through any saddles.
Abstract: This is a continuation of an earlier paper in which we investigated the superasymptotics and hyperasymptotics of the generalized Bessel function ϕ ( z ) = ∑ l = 0 ∞ z l Γ ( l + 1 ) Γ ( ρ l + β ) . where 0 < φ < X and g may be real or complex. In this paper, we consider the case -1 < φ < 0. The analysis in the two cases is not quite the same. Here we shall see that contributions towards hyperasymptotic expansions not only come from adjacent saddles but also from curves that are not even steepest-descent paths through any saddles.
29 citations
TL;DR: The zero of the Meixner polynomial is real, distinct, and lies in (0, ∞) as mentioned in this paper, where n is the number of vertices of the polynomials in n. For each fixed s, asymptotic formulas are obtained for both n,sand?n,s, asn?∞.
20 citations
16 citations
TL;DR: In this paper, the authors established the global existence and uniqueness of the solution u(x, t; ǫ) and proved its boundedness in x∈R and t>0 for all sufficiently small ǒ>0.
Abstract: Consider the nonlinear wave equation
utt−γ2uxx+f(u) = 0
with the initial conditions
u(x,0) = eφ(x), ut(x,0) = eψ(x),
where f(u) is either of the form f(u)=c2u−σu2s+1, s=1, 2,…, or an odd smooth function with f′(0)>0 and |f′(u)|≤C02.The initial data ϕ(x)∈C2 and ψ(x)∈C1 are odd periodic functions that have the same period. We establish the global existence and uniqueness of the solution u(x, t; ɛ), and prove its boundedness in x∈R and t>0 for all sufficiently small ɛ>0. Furthermore, we show that the error between the solution u(x, t; ɛ) and the leading term approximation obtained by the multiple scale method is of the order ɛ3 uniformly for x∈R and 0≤t≤T/ɛ2, as long as ɛ is sufficiently small, T being an arbitrary positive number.
3 citations