On the zeros of the Scorer functions
type: Item
instance of: scholarly article
title: On the zeros of the Scorer functions
zbMATH Open document ID: 1010.33001
DOI: 10.1016/S0021-9045(02)00022-9
published in: Journal of Approximation Theory
publication date: 2003-04-02T00:00:00Z
review text: The authors develop asymptotic approximations for the zeros of the functions Gi\((z)\) and Hi\((z)\), where \[ \text{ Gi}(z)=\frac{1}{\pi}\int_{0}^{\infty} \sin(zt+\frac{1}{3}t^{3}) dt \] and it is a particular solution of the differential equation \(w^{\prime \prime}-zw=-1/\pi\) and \[ \text{Hi} (z)=\frac{1}{\pi}\int_{0}^{\infty} e^{(zt-\frac{1}{3}t^{3})} dt \] and it is a solution of \(w^{\prime \prime}-zw=1/\pi\). As it is known \(w^{\prime \prime}-zw= \pm 1/\pi\), is the non-homogeneous Airy differential equation and the functions Gi\((z)\) and Hi\((z)\) are called the Scorer functions. The authors study qualitative properties of the real zeros of Gi\((z)\) and \(\text{ Gi}'(z)\) as well as asymptotics of the negative zeros of Gi\((z)\). They also study the complex zeros of Gi\((z)\), \(\text{Gi}'(z)\) and Hi\((z)\). Tables are given with numerical values of the zeros.
reviewed by: Stamatis Koumandos
Mathematics Subject Classification ID: 33C10
zbMATH DE Number: 1891286
zbMATH Keywords: non-homogeneous Airy differential equation
zbMATH Keywords: properties of the zeros of the solutions
zbMATH Keywords: Scorer functions
author: Amparo Gil
author: Javier Segura
author: Nico M. Temme
MaRDI profile type: Publication
cites work: On nonoscillating integrals for computing inhomogeneous Airy functions
cites work: Algorithm 822: GIZ, HIZ
cites work: Q3126395
full work available at URL: s0021-9045(02)00022-9
OpenAlex ID: W2044379363
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description: scientific article; zbMATH DE number 1891286