arXiv subjects

T. M. Dunster

Publications and source records attributed to T. M. Dunster.

At least 19 recordsLinked to original sources

arXiv2026

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

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 polynomials. The results are uniformly valid for $0\leqμ\leq(1-δ_0)κ<κ$, where $δ_0\in(0,1)$ is arbitrary and fixed. The analysis is based on the associated differential equation, which has a double pole and two turning points. At one of the turning points, which may coalesce with the double pole, a recently developed asymptotic theory is applied to obtain expansions involving Bessel functions. At the second turning point, expansions involving Airy functions are obtained. In both cases, the coefficients are readily computable, in contrast to those occurring in earlier results. The expansions, when taken together, uniformly cover the entire complex $z$-plane on the principal branch. Standard analytic continuation and connection formulas extend the results to all branches of $z$ and to negative $μ$.

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arXiv2026

The general Brannan coefficient conjecture I: Watson-lemma approximations

The coefficients $A_n(α,β,ω)$ in the Maclaurin expansion $(1+ωz)^α(1-z)^{-β}= \sum_{n=0}^{\infty} A_n(α,β,ω)z^n$ are studied, where $ω,z \in \mathbb{C}$ with $|z| < |ω|=1$, and $α,β\in (0,1]$. In 1973 Brannan conjectured that $|A_n(α,β,ω)|\le A_n(α,β,1)$ for each positive odd integer $n$, and showed it is true for $n=3$. This has recently been proven for all odd integers $n\ge5$ by a number of authors in aggregate for the special case $β=1$. In this paper hypergeometric integral representations and Watson-type approximations are utilised, from which the general problem is reduced to numerically evaluating the minima of certain simple, explicit, slowly-varying functions over compact domains. From the positivity of these constants it is shown that the conjecture holds for $α, β\in (0,1]$, $0 \le |\arg(ω)| \le π-ϕ_0$ and $n=5,7,9,\ldots$, where $ϕ_0=0.061$.

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arXiv2026

The general Brannan coefficient conjecture II: Meijer-function approximations

The coefficients $A_n(α,β,ω)$ in the Maclaurin expansion $(1+ωz)^α(1-z)^{-β}=\sum_{n=0}^{\infty} A_n(α,β,ω)z^n$ are considered for $|ω|=1$ and $α,β\in(0,1]$. D. A. Brannan conjectured in a 1973 paper that $|A_n(α,β,ω)|\le A_n(α,β,1)$ for every positive odd integer $n$. The present author recently established the conjecture outside a small neighbourhood of $ω=-1$. The remaining range is treated here by combining compound Laplace integral representations with two types of local approximation: a Meijer $G$ function approximation for $n|\arg(-ω)|$ bounded, and a modified Watson approximation for the complementary range. The resulting lower bounds reduce the problem to numerical positivity checks for explicit functions on compact parameter sets. These computations verify the inequality for all $α,β\in(0,1]$ and all odd integers $n\ge5$, and hence, together with Brannan's result for $n=3$, complete the proof of his conjecture.

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arXiv2025

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

The asymptotic behavior of solutions to the second-order linear differential equation $d^{2}w/dz^{2}=\{u^{2}f(α,z)+g(z)\}w$ is analyzed for a large real parameter $u$ and $α\in[0,α_{0}]$, where $α_{0}>0$ is fixed. The independent variable $z$ ranges over a complex domain $Z$ (possibly unbounded) on which $f(α,z)$ and $g(z)$ are analytic except at $z=0$, where the differential equation has a regular singular point. For $α>0$, the function $f(α,z)$ has a double pole at $z=0$ and a simple zero in $Z$, and as $α\to 0$ the turning point coalesces with the pole. Bessel function approximations are constructed for large $u$ involving asymptotic expansions that are uniformly valid for $z\in Z$ and $α\in[0,α_{0}]$. The expansion coefficients are generated by simple recursions, and explicit error bounds are obtained that simplify earlier results. As an application, uniform asymptotic expansions are derived for associated Legendre functions of large degree $ν$, valid for complex $z$ in an unbounded domain and for order $μ\in[0,ν(1-δ)]$, where $δ>0$ is arbitrary.

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arXiv2025

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

Linear second-order ordinary differential equations of the form $d^{2}w/dz^{2}=\{u^{2}f(a,z)$ $+g(z)\}w$ are studied for large values of the real parameter $u$, where $z$ ranges over a bounded or unbounded complex domain $Z$, and $a_{0} \le a \le a_{1} < \infty$. The functions $f(a,z)$ and $g(z)$ are analytic in the interior of $Z$. Moreover, $f(a,z)$ has exactly two real simple zeros in $Z$ for $a>a_{0}$ that depend continuously on $a$ and coalesce into a double zero as $a \to a_{0}$. Uniform asymptotic expansions are obtained for solutions in terms of parabolic cylinder functions and their derivatives, together with slowly varying coefficient functions. The coefficients are readily computable and explicit error bounds are provided. The results are then applied to derive new asymptotic expansions for the associated Legendre functions when both the degree $ν$ and the order $μ$ are large.

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arXiv2025

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

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 $-Δ_{1} n+\frac{3}{2} \leq a \leq Δ_{2} n$ for fixed arbitrary $Δ_{1} \in (0,1)$ and bounded positive $Δ_{2}$. For this parameter range at most one of the zeros is real, with the rest being complex conjugates. The new expansions are uniformly valid for all the zeros, and are shown to be highly accurate for moderate or large values of $n$. They are consequently used as initial values in a very efficient numerical algorithm designed to obtain the remaining complex zeros using Taylor series.

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arXiv2025

Uniform asymptotic expansions for generalised trigonometric integrals and their zeros

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 complex values of the argument. These follow from new Liouville-Green asymptotic expansions for incomplete gamma functions. Asymptotic expansions for the real zeros of the generalised trigonometric integrals are then constructed for large $a$ which are uniformly valid without restriction on their size (small or large).

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arXiv2025

Simplified Airy function Asymptotic expansions for Reverse Generalised Bessel Polynomials

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 previously known results. The defining differential equation is analysed; for large $n$ and $\frac{3}{2} - n < a < \infty$, it possesses two turning points in the complex $z$ plane which are complex conjugates. Away from these turning points Liouville-Green expansions are obtained for the polynomials and two companion solutions of the differential equation, where asymptotic series appear in the exponent. Then representations involving Airy functions and two slowly varying coefficient functions are constructed. Using the Liouville-Green representations, asymptotic expansions are obtained for the coefficient functions that involve coefficients that can be easily and explicitly computed recursively. In conjunction with a suitable re-expansion, or Cauchy's integral formula, near the turning point, the expansions are valid for $-Δ_{1} n+\frac{3}{2} \leq a \leq Δ_{2} n$ for fixed arbitrary $Δ_{1} \in (0,1)$ and bounded positive $Δ_{2}$, uniformly for all unbounded complex values of $z$.

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arXiv2025

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

Asymptotic expansions are derived for Gegenbauer (ultraspherical) polynomials for large order $n$ that are uniformly valid for unbounded complex values of the argument $z$, including the real interval $0 \leq z \leq 1$ in which the zeros in the right half plane are located: symmetry extends the results to the left half plane. The approximations are derived from the differential equation satisfied by these polynomials, and other independent solutions are also considered. For large $n$ this equation is characterized by having a simple pole, and expansions valid at this singularity involve Bessel functions and slowly varying coefficient functions. The expansions for these functions are simpler than previous approximations, in particular being computable to a high degree of accuracy. Simple explicit error bounds are derived which only involve elementary functions, and thereby provide a simplification of previous expansions and error bounds associated with differential equations having a large parameter and simple pole.

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arXiv2025

Simplified uniform asymptotic expansions for associated Legendre and conical functions

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 valid for unbounded real and complex values of the argument $z$, including the singularity $z=1$. The cases where $ν+\frac12$ and $μ$ are real or purely imaginary are included, which covers conical functions. The approximations involve either exponential or modified Bessel functions, along with slowly varying coefficient functions. The coefficients of the new asymptotic expansions are simple and readily obtained explicitly, allowing for computation to a high degree of accuracy. The results are constructed and rigorously established by employing certain Liouville-Green type expansions where the coefficients appear in the exponent of an exponential function.

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arXiv2025

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

Bessel and modified Bessel functions of imaginary order $iν$ ($ν>0$) are studied. Asymptotic expansions are derived as $ν\to \infty$ that are uniformly valid in unbounded complex domains, with error bounds provided. Coupled with appropriate connection formulas the approximations are uniformly valid for all complex argument. The expansions are of two forms, Liouville-Green type expansions only involving elementary functions, and ones involving Airy functions that are valid at a turning point of the defining differential equation. The new results have coefficients and error bounds that are simpler than in prior expansions, and further are used to construct asymptotic expansions for the zeros of Bessel and modified Bessel functions of large imaginary order, these being uniformly valid without restriction on their size (small or large).

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arXiv2025

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

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 accurate approximations to the first zero plus a highly efficient method based on a fourth-order fixed point method with the parabolic cylinder functions computed by Taylor series and carefully selected steps, to compute the rest of the zeros. For $|a|$ small, the asymptotic approximations are complemented with a few fixed point iterations requiring the evaluation of $U(a,z)$ and $U'(a,z)$ in the region where the complex zeros are located. Liouville-Green expansions are derived to enhance the performance of a computational scheme to evaluate $U(a,z)$ and $U'(a,z)$ in that region. Several tests show the accuracy and efficiency of the numerical algorithm.

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arXiv2025

Error bounds for a uniform asymptotic approximation of the zeros of the Bessel function $J_ν(x)$

A recent asymptotic expansion for the positive zeros $x=j_{ν,m}$ ($m=1,2,3,\ldots$) of the Bessel function of the first kind $J_ν(x)$ is studied, where the order $ν$ is positive. Unlike previous well-known expansions in the literature, this is uniformly valid for one or both $m$ and $ν$ unbounded, namely $m=1,2,3,\ldots$ and $1 \leq ν< \infty$. Explicit and simple lower and upper error bounds are derived for the difference between $j_{ν,m}$ and the first three terms of the expansion. The bounds are sharp in the sense they are close to the value of the fourth term of the expansion (i.e. the first neglected term).

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arXiv2024

Uniform asymptotic expansions for the zeros of parabolic cylinder functions

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 these are valid for $a$ positive or negative and large in absolute value, uniformly for unbounded $z$ (real or complex). The accuracy of the approximations of the complex zeros is then demonstrated with some comparative tests using a highly precise numerical algorithm for finding the complex zeros of the function.

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arXiv2024

Uniform asymptotic expansions for the zeros of Bessel functions

Reformulated uniform asymptotic expansions are derived for ordinary differential equations having a large parameter and a simple turning point. These involve Airy functions, but not their derivatives, unlike traditional asymptotic expansions. From these, asymptotic expansions are derived for the zeros of Bessel functions that are valid for large positive values of the order, uniformly valid for all the zeros. The coefficients in the expansions are explicitly given elementary functions, and similar expansions are derived for the zeros of the derivatives of Bessel functions.

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arXiv2022

Computation of parabolic cylinder functions having complex argument

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 exponential and Airy functions, with slowly varying analytic coefficient functions involving simple coefficients, and stable integral representations; these two main main methods can be complemented with Maclaurin series and a Poincaré asymptotic expansion. We provide numerical evidence showing that the combination of these methods is enough for computing the function with $5\times 10^{-13}$ relative accuracy in double precision floating point arithmetic.

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arXiv2021

Nield-Kuznetsov functions and Laplace transforms of parabolic cylinder functions

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 formulas are constructed between them, numerically satisfactory solutions of the homogeneous version of the differential equation, and a new complementary Nield-Kuznetsov function. Asymptotic expansions are then derived that are uniformly valid for large values of the parameter and unbounded real and complex values of the argument. Laplace transforms of the parabolic cylinder functions $W(a,x)$ and $U(a,x)$ are subsequently shown to be explicitly represented in terms of the complementary Nield-Kuznetsov function and closely related functions, and from these uniform asymptotic expansions are derived for the integrals.

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arXiv2021

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

The function $g(x)= (1+1/x)^{x}$ has the well-known limit $e$ as $x\rightarrow{\infty}$. The coefficients $c_{j}$ in an asymptotic expansion for $g(x)$ are considered. A simple recursion formula is derived, and then using Cauchy's integral formula the coefficients are approximated for large $j$. From this it is shown that $|c_{j}|\rightarrow{1}$ as $j\rightarrow{\infty}$.

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