T. M. Dunster

activity

20142026

most citedFundamental Solutions and Gegenbauer Expansions of Helmholtz Operators in Riemannian Spaces of Constant Curvature

14 citations · 20 across the 31 of their papers we have counts for

collaborators

32 papers

math.CA2026

Whittaker functions with one or both parameters large: simplified uniform asymptotic expansions involving Bessel and Airy functions

T. M. Dunster

Uniform asymptotic expansions are derived as $κ \to \infty$ for the Whittaker functions $W_{κ, μ} (z)$ , $M_{κ, μ} (z)$ , as well as related functions including generalized Laguerre polyn…

math.CA2026

The general Brannan coefficient conjecture II: Meijer-function approximations

T. M. Dunster

The coefficients $A_{n} (α, β, ω)$ in the Maclaurin expansion $(1 + ω z)^{α} (1 - z)^{- β} = \sum_{n = 0}^{\infty} A_{n} (α, β, ω) z^{n}$ are considered for $∣ ω ∣ = 1$ and $α, β \in (0, 1]$ . D. A. Brannan conjectu…

math.CA2026

The general Brannan coefficient conjecture I: Watson-lemma approximations

T. M. Dunster

The coefficients $A_{n} (α, β, ω)$ in the Maclaurin expansion $(1 + ω z)^{α} (1 - z)^{- β} = \sum_{n = 0}^{\infty} A_{n} (α, β, ω) z^{n}$ are studied, where $ω, z \in C$ with $∣ z ∣ < ∣ ω ∣ = 1$ , and $α,…

math.CA2025

Uniform Asymptotic approximation and numerical evaluation of the Reverse Generalized Bessel Polynomial zeros

T. M. Dunster, Amparo Gil, Diego Ruiz-Antolin +1

Uniform asymptotic expansions are derived for the zeros of the reverse generalized Bessel polynomials of large degree $n$ and real parameter $a$ . It is assumed that $-Δ_{1} n+\frac…

math.CA2025

Asymptotic expansions for solutions of differential equations having a coalescing turning point and double pole, with an application to Legendre functions

T. M. Dunster

The asymptotic behavior of solutions to the second-order linear differential equation $d^{2} w / d z^{2} = {u^{2} f (α, z) + g (z)} w$ is analyzed for a large real parameter $u$ and $α\in[0,α…

math.CA2025

Uniform asymptotic expansions for generalised trigonometric integrals and their zeros

T. M. Dunster

Asymptotic expansions for generalised trigonometric integrals are obtained in terms of elementary functions, which are valid for large values of the parameter $a$ and unbounded com…

math.CA2025

Simplified Airy function Asymptotic expansions for Reverse Generalised Bessel Polynomials

T. M. Dunster

Uniform asymptotic expansions are derived for reverse generalised Bessel polynomials of large degree $n$ , real parameter $a$ , and complex argument $z$ , which are simpler than previ…

math.CA2025

Asymptotic expansions for solutions of differential equations having coalescing turning points, with an application to Legendre functions

T. M. Dunster

Linear second-order ordinary differential equations of the form $d^{2} w / d z^{2} = {u^{2} f (a, z)$ $+ g (z)} w$ are studied for large values of the real parameter $u$ , where $z$ ranges ov…

math.NA2024

A numerical algorithm for computing the zeros of parabolic cylinder functions in the complex plane

T. M. Dunster, A. Gil, D. Ruiz-Antolín +1

A numerical algorithm (implemented in Matlab) for computing the zeros of the parabolic cylinder function $U (a, z)$ in domains of the complex plane is presented. The algorithm uses a…

math.CA2024

Uniform asymptotic expansions for Bessel functions of imaginary order and their zeros

T. M. Dunster

Bessel and modified Bessel functions of imaginary order $i ν$ ( $ν > 0$ ) are studied. Asymptotic expansions are derived as $ν \to \infty$ that are uniformly valid in unbounded complex d…

math.CA2024

Simplified uniform asymptotic expansions for associated Legendre and conical functions

T. M. Dunster

Asymptotic expansions are derived for associated Legendre functions of degree $ν$ and order $μ$ , where one or the other of the parameters is large. The expansions are uniformly val…

math.CA2024

Uniform asymptotic expansions for the zeros of parabolic cylinder functions

T. M. Dunster, A. Gil, D. Ruiz-Antolin +1

The real and complex zeros of the parabolic cylinder function $U (a, z)$ are studied. Asymptotic expansions for the zeros are derived, involving the zeros of Airy functions, and thes…

math.CA2024

Error bounds for a uniform asymptotic approximation of the zeros of the Bessel function $J_{ν} (x)$

T. M. Dunster

A recent asymptotic expansion for the positive zeros $x = j_{ν, m}$ ( $m = 1, 2, 3, \dots$ ) of the Bessel function of the first kind $J_{ν} (x)$ is studied, where the order $ν$ is positive. U…

math.CA2023

Uniform asymptotic expansions for the zeros of Bessel functions

T. M. Dunster

Reformulated uniform asymptotic expansions are derived for ordinary differential equations having a large parameter and a simple turning point. These involve Airy functions, but no…

math.NA2022

Computation of parabolic cylinder functions having complex argument

T. M. Dunster, A. Gil, J. Segura

Numerical methods for the computation of the parabolic cylinder $U (a, z)$ for real $a$ and complex $z$ are presented. The main tools are recent asymptotic expansions involving expon…

math.CA2022

Uniform asymptotic expansions for Gegenbauer polynomials and related functions via differential equations having a simple pole

T. M. Dunster

Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order $n$ that are uniformly valid for unbounded complex values of the argument $z$ , includi…

math.CA2021★ 1 cited

Uniform asymptotic expansions for the Whittaker functions $M_{κ, μ} (z)$ and $W_{κ, μ} (z)$ with $μ$ large

T. M. Dunster

Uniform asymptotic expansions are derived for Whittaker's confluent hypergeometric functions $M_{κ, μ} (z)$ and $W_{κ, μ} (z)$ , as well as the numerically satisfactory companion functi…

math.CA2021

On the coefficients in an asymptotic expansion of $(1 + 1/ x)^{x}$

T. M. Dunster, Jessica M. Perez

The function $g (x) = (1 + 1/ x)^{x}$ has the well-known limit $e$ as $x \to \infty$ . The coefficients $c_{j}$ in an asymptotic expansion for $g (x)$ are considered. A simple rec…

math.CA2021

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

T. M. Dunster

Using a differential equation approach asymptotic expansions are rigorously obtained for Lommel, Weber, Anger-Weber and Struve functions, as well as Neumann polynomials, each of wh…

math.CA2021

Nield-Kuznetsov functions and Laplace transforms of parabolic cylinder functions

T. M. Dunster

Nield-Kuznetsov functions of the first kind are studied, which are solutions of an inhomogeneous parabolic Weber equation, and have applications in fluid flow problems. Connection…

math.CA2020

Uniform asymptotic expansions for solutions of the parabolic cylinder and Weber equations

T. M. Dunster

Asymptotic expansions are derived for solutions of the parabolic cylinder and Weber differential equations. In addition the inhomogeneous versions of the equations are considered,…

math.CA2020

Sharp error bounds for turning point expansions

T. M. Dunster, A. Gil, J. Segura

Computable and sharp error bounds are derived for asymptotic expansions for linear differential equations having a simple turning point. The expansions involve Airy functions and s…

math.CA2020

Asymptotic solutions of inhomogeneous differential equations having a turning point

T. M. Dunster

Asymptotic solutions are derived for inhomogeneous differential equations having a large real or complex parameter and a simple turning point. They involve Scorer functions and thr…

math.CA2019

Simplified error bounds for turning point expansions

T. M. Dunster, A. Gil, J. Segura

Recently, the present authors derived new asymptotic expansions for linear differential equations having a simple turning point. These involve Airy functions and slowly varying coe…

Whittaker functions with one or both parameters large: simplified uniform asymptotic expansions involving Bessel and Airy functions

The general Brannan coefficient conjecture II: Meijer-function approximations

The general Brannan coefficient conjecture I: Watson-lemma approximations

Uniform Asymptotic approximation and numerical evaluation of the Reverse Generalized Bessel Polynomial zeros

Asymptotic expansions for solutions of differential equations having a coalescing turning point and double pole, with an application to Legendre functions

Uniform asymptotic expansions for generalised trigonometric integrals and their zeros

Simplified Airy function Asymptotic expansions for Reverse Generalised Bessel Polynomials

Asymptotic expansions for solutions of differential equations having coalescing turning points, with an application to Legendre functions

A numerical algorithm for computing the zeros of parabolic cylinder functions in the complex plane

Uniform asymptotic expansions for Bessel functions of imaginary order and their zeros

Simplified uniform asymptotic expansions for associated Legendre and conical functions

Uniform asymptotic expansions for the zeros of parabolic cylinder functions

Error bounds for a uniform asymptotic approximation of the zeros of the Bessel function Jν​(x)

Uniform asymptotic expansions for the zeros of Bessel functions

Computation of parabolic cylinder functions having complex argument

Uniform asymptotic expansions for Gegenbauer polynomials and related functions via differential equations having a simple pole

Uniform asymptotic expansions for the Whittaker functions Mκ,μ​(z) and Wκ,μ​(z) with μ large

On the coefficients in an asymptotic expansion of (1+1/x)x

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

Nield-Kuznetsov functions and Laplace transforms of parabolic cylinder functions

Uniform asymptotic expansions for solutions of the parabolic cylinder and Weber equations

Sharp error bounds for turning point expansions

Asymptotic solutions of inhomogeneous differential equations having a turning point

Simplified error bounds for turning point expansions

Error bounds for a uniform asymptotic approximation of the zeros of the Bessel function $J_{ν} (x)$

Uniform asymptotic expansions for the Whittaker functions $M_{κ, μ} (z)$ and $W_{κ, μ} (z)$ with $μ$ large

On the coefficients in an asymptotic expansion of $(1 + 1/ x)^{x}$