Generalized Bessel functions of the first kind
type: Item
instance of: scholarly article
title: Generalized Bessel functions of the first kind
zbMATH Open document ID: 1203.33001
DOI: 10.1007/978-3-642-12230-9
author: Árpád Baricz
published in: Lecture Notes in Mathematics
publication date: 2010-05-27T00:00:00Z
review text: This book deals with properties of generalized Bessel functions of order \(p\), which are defined as solutions of the differential equation \[ z^2w''(z)+ bz w'(z)+ [cz^2- p^2+(1- b)p]w(z)= 0, \] which in case \(b= c=1\) reduces to the (ordinary) Bessel differential equation. Moreover by specialization of the constants \(b\) and \(c\) one obtains the differential equation for the modified Bessel and for the spherical Bessel functions. So all these functions can be considered in a unified manner. In particular the function \[ w_p(z)= \sum^\infty_{n=0} {(-c)^n\over n!\Gamma[p+ n+ (b+ 1)/2]} \biggl({z\over 2}\biggr)^{2n+p},\quad z\in\mathbb{C} \] is called generalized Bessel function of the first kind of order \(p\). The author writes: ``In this book our aim is to study the generalized Bessel functions of the first kind from the point of view of the complex analysis on the one hand, and on the other hand from the point of view of the classical analysis.'' This leads on the one hand to results related to univalence, starlikeness, convexity and others and on the other hand to results on the extension of trigonometric inequalities like Mitrinovic, Matragan, Redheffer, Cusa, Jordan, Grünbaum, Landen, van der Corput and Askey, to generalized Bessel functions of the first kind. Chapter 1 deals with the transfer for some analytical properties of the (ordinary) Bessel functions of the first kind to their generalization. In Chapter 2 geometric properties of generalized Bessel functions as univalence, starlikeness and convexity properties are derived. The last Chapter 3 contains inequalities as indicated above, many taken from original papers of the author. In an Appendix, ``Conjectures'' and ``Open problems'', formulated by the author and co-authors of original papers are stated as \texttt{Matlab} programs for graphs. The References contain 246 items, 27 of them are papers of the author partly with co-authors. To my knowledge, the results are presented in a monograph for the fir…
Mathematics Subject Classification ID: 33-02
Mathematics Subject Classification ID: 33C10
Mathematics Subject Classification ID: 30C45
Mathematics Subject Classification ID: 26D07
zbMATH DE Number: 5712790
zbMATH Keywords: Bessel functions
zbMATH Keywords: generalized Bessel functions
zbMATH Keywords: starlikeness
zbMATH Keywords: convexity
zbMATH Keywords: inequalities
Wikidata QID: Q121027903
reviewed by: Hans-Jürgen Glaeske
describes a project that uses: Matlab
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/978-3-642-12230-9
OpenAlex ID: W82553446
description: scientific article; zbMATH DE number 5712790