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The radius of starlikeness of regular Coulomb wave functions

Árpád Baricz, Pranav Kumar, Sanjeev Singh

TL;DR

This work determines radii of univalence and radii of starlikeness for two normalised regular Coulomb wave functions, and introduces a generalized normalised Bessel function to extend geometric-function-theory techniques to Coulomb-type functions. It leverages Brown's differential-inequality framework and Rayleigh-sum expansions of Coulomb zeros to obtain exact radii as smallest modulus roots of derivative-based equations, and derives a complete asymptotic expansion for the radius of starlikeness in the large-order limit with a recursive scheme for the coefficients. The results connect Coulomb wave functions with Bessel-function analogues, yielding corollaries for related normalised forms and broadening the understanding of univalence and starlikeness in special-function contexts. A key contribution is the explicit asymptotic expansion $r^{*}(f_{L,\eta})=L\left(\sqrt{2}+\left(\sqrt{2}\eta+\frac{1}{2\sqrt{2}}-\frac{1}{2}\right)\frac{1}{L}+\cdots\right)$, together with a robust recurrence to compute higher-order terms.

Abstract

Motivated by the pioneering work of M.S. Robertson [Ro54] and R.K. Brown [Br60], [Br62], who examined the geometric properties of some normalised solutions of second-order homogeneous differential equations, in this paper we investigate the radii of univalence and starlikeness for two kind of normalised regular Coulomb wave functions. Moreover, a generalized normalised Bessel function is introduced, and its radius of starlikeness is studied by using two different approaches. In addition, the asymptotic behaviour, with respect to the large order, of the radius of starlikeness of one type of normalised Coulomb wave functions is considered, which is in fact the first zero of the derivative of the regular Coulomb wave function. We derive a complete asymptotic expansion for this radius of starlikeness and provide a recurrence relation for the coefficients of this expansion. The proof is based on Rayleigh sums of the zeros of Coulomb wave functions, asymptotic inversion and some basic results on regular Coulomb wave functions developed by Štampach and Štovíček [SS14].

The radius of starlikeness of regular Coulomb wave functions

TL;DR

This work determines radii of univalence and radii of starlikeness for two normalised regular Coulomb wave functions, and introduces a generalized normalised Bessel function to extend geometric-function-theory techniques to Coulomb-type functions. It leverages Brown's differential-inequality framework and Rayleigh-sum expansions of Coulomb zeros to obtain exact radii as smallest modulus roots of derivative-based equations, and derives a complete asymptotic expansion for the radius of starlikeness in the large-order limit with a recursive scheme for the coefficients. The results connect Coulomb wave functions with Bessel-function analogues, yielding corollaries for related normalised forms and broadening the understanding of univalence and starlikeness in special-function contexts. A key contribution is the explicit asymptotic expansion , together with a robust recurrence to compute higher-order terms.

Abstract

Motivated by the pioneering work of M.S. Robertson [Ro54] and R.K. Brown [Br60], [Br62], who examined the geometric properties of some normalised solutions of second-order homogeneous differential equations, in this paper we investigate the radii of univalence and starlikeness for two kind of normalised regular Coulomb wave functions. Moreover, a generalized normalised Bessel function is introduced, and its radius of starlikeness is studied by using two different approaches. In addition, the asymptotic behaviour, with respect to the large order, of the radius of starlikeness of one type of normalised Coulomb wave functions is considered, which is in fact the first zero of the derivative of the regular Coulomb wave function. We derive a complete asymptotic expansion for this radius of starlikeness and provide a recurrence relation for the coefficients of this expansion. The proof is based on Rayleigh sums of the zeros of Coulomb wave functions, asymptotic inversion and some basic results on regular Coulomb wave functions developed by Štampach and Štovíček [SS14].
Paper Structure (4 sections, 14 theorems, 173 equations, 2 figures)

This paper contains 4 sections, 14 theorems, 173 equations, 2 figures.

Table of Contents

  1. Introduction and Preliminaries
  2. The radii of univalence and starlikeness of Coulomb wave functions
  3. Asymptotics of the radius of starlikeness of the normalised Coulomb wave function
  4. Proofs of the main results

Key Result

Theorem 1

Brown Let $z^2p(z)$, defined as in p_exp, be regular for $|z|<r$ and satisfy the inequality where $c\geq0,$$|\gamma|\leq \pi/2$ and $z^2p^*(z)$ is defined in p*. With $p(z)$ chosen in this manner we define to be the unique solution of int_de_1 for $|z|<r$ corresponding to the root with larger real part of the associated characteristic equation. Let $W_c(z)$ be defined as in W_c. Then for all $|

Figures (2)

  • Figure 1: The image of the open disk $\mathbb{D}_{\widetilde{\rho}_{0,0,1}}$ under the Bessel function $z\mapsto f_{-\frac{1}{2},0}(z)=\left[\sqrt{z}J_0(z)\right]^2,$ where $\widetilde{\rho}_{0,0,1}\sim 0.9407705639497375\ldots$ is the smallest positive zero of the function $r\mapsto 2rJ_0'(r)+J_0(r).$
  • Figure 2: The image of the open disk $\mathbb{D}_{\rho^*_{\frac{1}{2},0,1}}$ under the trigonometric function $z\mapsto g_{0,0}(z)=\sqrt{\frac{\pi z}{2}}J_{\frac{1}{2}}(z)=\sin z,$ where $\rho^*_{\frac{1}{2},0,1}\sim 1.5707963267948968\ldots$ is the smallest positive zero of the function $r\mapsto 2rJ_{\frac{1}{2}}'(r)+J_{\frac{1}{2}}(r).$

Theorems & Definitions (24)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Corollary 1
  • Corollary 2
  • Theorem 3
  • Corollary 3
  • Theorem 4
  • Corollary 4
  • Corollary 5
  • ...and 14 more