Lommel Functions Research Articles

Radii of Uniform Convexity of Lommel and Struve Functions

In this paper, we determine the radii of $$\beta $$ -uniform convexity of order $$\alpha $$ for six kinds of normalized Lommel and Struve functions of the first kind. One of the most important things which we have learned in this study is that the radii of uniform convexity are obtained as solutions of some transcendental equations.

On Hurwitz zeta function and Lommel functions

We obtain a new proof of Hurwitz’s formula for the Hurwitz zeta function [Formula: see text] beginning with Hermite’s formula. The aim is to reveal a nice connection between [Formula: see text] and a special case of the Lommel function [Formula: see text]. This connection is used to rephrase a modular-type transformation involving infinite series of Hurwitz zeta function in terms of those involving Lommel functions.

Radii of Starlikeness and Convexity of Some Entire Functions

A normalized analytic function f is lemniscate starlike if the quantity $$zf'(z)/f(z)$$ lies in the region bounded by the right half of the lemniscate of Bernoulli $$|w^2-1|=1$$ . It is Janowski starlike if the quantity $$zf'(z)/f(z)$$ lies in the disk whose diametric end points are $$(1-A)/(1-B)$$ and $$(1+A)/(1+B)$$ for $$-1\le B<A\le 1$$ . The radii of lemniscate starlikeness and Janowski starlikeness have been determined for normalizations of q-Bessel functions, Bessel functions of first kind of order $$\nu $$ and Lommel functions of first kind. Corresponding convexity radii are also determined.

Bounds for modified Lommel functions of the first kind and their ratios

The modified Lommel function tμ,ν(x) is an important special function, but to date there has been little progress on the problem of obtaining functional inequalities for tμ,ν(x). In this paper, we advance the literature substantially by obtaining a simple two-sided inequality for the ratio tμ,ν(x)/tμ−1,ν−1(x) in terms of the ratio Iν(x)/Iν−1(x) of modified Bessel functions of the first kind, thereby allowing one to exploit the extensive literature on bounds for this ratio. We apply this result to obtain two-sided inequalities for the condition numbers xtμ,ν′(x)/tμ,ν(x), the ratio tμ,ν(x)/tμ,ν(y) and the modified Lommel function tμ,ν(x) itself that are given in terms of xIν′(x)/Iν(x), Iν(x)/Iν(y) and Iν(x), respectively. The bounds obtained in this paper are quite accurate and often tight in certain limits. As an important special case we deduce bounds for modified Struve functions of the first kind and their ratios, some of which are new, whilst others extend the range of validity of some results given in the recent literature.

Inequalities for Some Integrals Involving Modified Lommel Functions of the First Kind

In this paper, we obtain inequalities for some integrals involving the modified Lommel function of the first kind t_{mu ,nu }(x). In most cases, these inequalities are tight in certain limits. We also deduce a tight double inequality, involving the modified Lommel function t_{mu ,nu }(x), for a generalized hypergeometric function. The inequalities obtained in this paper generalise recent bounds for integrals involving the modified Struve function of the first kind.

On the curves of zeros of Lommel functions of two variables

ABSTRACTIn this paper, we study some properties of the locus of zeros of the Lommel function of two variables . Furthermore, we find the regions on which the curves of zeros of lie, when ν is large.

Certain Geometric Properties of Lommel and Hyper-Bessel Functions

In this article, we are mainly interested in finding the sufficient conditions under which Lommel functions and hyper-Bessel functions are close-to-convex with respect to the certain starlike functions. Strongly starlikeness and convexity of Lommel functions and hyper-Bessel functions are also discussed. Some applications are also the part of our investigation.

On the positivity and zeros of Lommel functions: Hyperbolic extension and interlacing

We obtain for the Lommel functions of the first kind a hyperbolic region of positivity, thereby extending earlier results of R.G. Cooke and J. Steinig. In addition, we prove that the zeros of Lommel functions for a large class of parameters are interlaced with those of Bessel functions.

Geometric Properties of Lommel Functions of the First Kind

In the present paper, we find sufficient conditions for starlikeness and convexity of normalized Lommel functions of the first kind using the admissible function methods. Additionally, we investigate some inclusion relationships for various classes associated with the Lommel functions. The functions belonging to these classes are related to the starlike functions, convex functions, close-to-convex functions and quasi-convex functions.

Modeling and analysis of multilayer piezoelectric-elastic spherical transducers

An exact analytical model of multilayer piezoelectric-elastic spherical transducers is established in this article, and their dynamic characteristics are studied. Based on the linear theory of piezoelasticity, the dynamic analytical solution is derived in terms of Bessel functions, Lommel functions, Gamma functions, and Generalized Hypergeometric functions, and the electric impedance is obtained to determine the resonance frequencies. By proper selection of the layer number and geometric dimensions, the proposed model can approximately degrade into some special cases in the previous works, including single-layer piezoelectric spherical transducer, two-layer piezoelectric-elastic spherical transducer, and three-layer sandwiched piezoelectric-elastic-piezoelectric spherical transducer. Comparisons between the results of the proposed model and the ones of these special cases are also given and good agreements are found. Furthermore, parametric studies are conducted to discuss the effects of the key parameters on the dynamic characteristics of these special cases and a multilayer case, such as the first resonance and anti-resonance frequencies as well as the corresponding electromechanical coupling factor. This work contributes to an overall understanding of the dynamic characteristics of miniature piezoelectric hollow sphere transducers (BBs) and stacked piezoelectric spherical transducers. The developed model can be used to guide the design of the multilayer piezoelectric-elastic spherical transducers for promising applications in underwater acoustic detection, ultrasonic imaging, hydrophones, and nondestructive testing.

Error Bounds for the Large‐Argument Asymptotic Expansions of the Lommel and Allied Functions

AbstractIn this paper, we reconsider the large‐z asymptotic expansion of the Lommel function and its derivative. New representations for the remainder terms of the asymptotic expansions are found and used to obtain sharp and realistic error bounds. We also give re‐expansions for these remainder terms and provide their error estimates. Applications to the asymptotic expansions of the Anger–Weber‐type functions, the Scorer functions, the Struve functions, and their derivatives are provided. The sharpness of our error bounds is discussed in detail, and numerical examples are given.

Two-sided inequalities for the Struve and Lommel functions

Mathematical inequalities and other results involving such widely- and extensively-studied special functions of mathematical physics and applied mathematics as (for example) the Bessel, Struve and Lommel functions as well as the associated hypergeometric functions are potentially useful in many seemingly diverse areas of applications, especially in situations in which these functions are involved in solutions of mathematical, physical and engineering problems which can be modeled by ordinary and partial differential equations. With this objective in view, our present investigation is motivated by some open problems involving inequalities for a number of particular forms of the hypergeometric function 1F2(a; b, c; z). Here, in this paper, we apply a novel approach to such problems and obtain presumably new two-sided inequalities for the Struve function, the associated Struve function and the modified Struve function by first investigating inequalities for the general hypergeometric function 1F2(a; b, c; z). We also briefly discuss the analogous new inequalities for the Lommel function under some conditions and constraints. Finally, as special cases of our main results, we deduce several inequalities for the modified Lommel function and the normalized Lommel function.

Summations of Schlömilch series containing anger function terms

Abstract The main aim of this article is to establish certain closed expressions for the Schlömilch series which members contain Anger function 𝐉 ν in order to generalize already known results. Closed expressions for the Schlömilch series with members containing some Lommel functions of the first kind s μ,ν and also a Weber function 𝐄 ν and Struve function 𝐇 ν are derived as a by–product of these results.

Bounds for radii of starlikeness and convexity of some special functions

In this paper we consider some normalized Bessel, Struve, and Lommel functions of the first kind and, by using the Euler--Rayleigh inequalities for the first positive zeros of a combination of special functions, we obtain tight lower and upper bounds for the radii of starlikeness of these functions. By considering two different normalizations of Bessel and Struve functions we give some inequalities for the radii of convexity of the same functions. On the other hand, we show that the radii of univalence of some normalized Struve and Lommel functions are exactly the radii of starlikeness of the same functions. In addition, by using some ideas of Ismail and Muldoon we present some new lower and upper bounds for the zeros of derivatives of some normalized Struve and Lommel functions. The Laguerre--Polya class of real entire functions plays an important role in our study.

Centroids analysis for circle of confusion in reverse Hartmann test**Project supported by the National Natural Science Foundation of China (Grant No. 61475018).

The point spread function (PSF) is investigated in order to study the centroids algorithm in a reverse Hartmann test (RHT) system. Instead of the diffractive Airy disk in previous researches, the intensity of PSF behaves as a circle of confusion (CoC) and is evaluated in terms of the Lommel function in this paper. The fitting of a single spot with the Gaussian profile to identify its centroid forms the basis of the proposed centroid algorithm. In the implementation process, gray compensation is performed to obtain an intensity distribution in the form of a two-dimensional (2D) Gauss function while the center of the peak is derived as a centroid value. The segmental fringe is also fitted row by row with the one-dimensional (1D) Gauss function and reconstituted by averaged parameter values. The condition used for the proposed method is determined by the strength of linear dependence evaluated by Pearson’s correlation coefficient between profiles of Airy disk and CoC. The accuracies of CoC fitting and centroid computation are theoretically and experimentally demonstrated by simulation and RHTs. The simulation results show that when the correlation coefficient value is more than 0.9999, the proposed centroid algorithm reduces the root-mean-square error (RMSE) by nearly one order of magnitude, thus achieving an accuracy of pixel or better performance in experiment. In addition, the 2D and 1D Gaussian fittings for the segmental fringe achieve almost the same centroid results, which further confirm the feasibility and advantage of the theory and method.

Bounds for the radii of univalence of some special functions

Tight lower and upper bounds for the radius of univalence of some normalized Bessel, Struve and Lommel functions of the first kind are obtained via Euler-Rayleigh inequalities. It is shown also that the radius of univalence of the Struve functions is greater than the corresponding radius of univalence of Bessel functions. Moreover, by using the idea of Kreyszig and Todd, and Wilf it is proved that the radii of univalence of some normalized Struve and Lommel functions are exactly the radii of starlikeness of the same functions. The Laguerre-Polya class of entire functions plays an important role in our study.

Bounds for the radii of univalence of some special functions

Tight lower and upper bounds for the radius of univalence of some normalized Bessel, Struve and Lommel functions of the first kind are obtained via Euler-Rayleigh inequalities. It is shown also that the radius of univalence of the Struve functions is greater than the corresponding radius of univalence of Bessel functions. Moreover, by using the idea of Kreyszig and Todd, and Wilf it is proved that the radii of univalence of some normalized Struve and Lommel functions are exactly the radii of starlikeness of the same functions. The Laguerre-Polya class of entire functions plays an important role in our study.

Positive Trigonometric Integrals Associated with Some Lommel Functions of the First Kind

We prove that $$\begin{aligned} \int _{0}^{x} t^{\mu -\frac{1}{2}}\,\log t\, \sin (x-t)\,\mathrm{d}t=\sqrt{x}\,\frac{\mathrm{d}}{\mathrm{d}\mu }\,s_{\mu ,\frac{1}{2}}(x)>0, \end{aligned}$$ for all \(\mu \ge 1/2\) and \(x\ge \pi \), where \(s_{\mu ,\frac{1}{2}}(x)\) is the Lommel function of the first kind. This refines a result of Steinig published in 1972. As an application, we derive positive functional bounds for the functions \(s_{\mu ,\frac{1}{2}}(x)\), \(\mu \ge 1/2\) and \(x\ge \pi \). Furthermore, we give estimates for the location of the zeros of the functions \(s_{\mu ,1/2}(x)\), \(-3/2 0\).

Centroids computation and point spread function analysis for reverse Hartmann test

This paper studies the point spread function (PSF) and centroids computation methods to improve the performance of reverse Hartmann test (RHT) in poor conditions, such as defocus, background noise, etc. In the RHT, we evaluate the PSF in terms of Lommel function and classify it as circle of confusion (CoC) instead of Airy disk. Approximation of a CoC spot with Gaussian or super-Gaussian profile to identify its centroid forms the basis of centroids algorithm. It is also effective for fringe pattern while the segmental fringe is served as a ‘spot’ with an infinite diameter in one direction. RHT experiments are conducted to test the fitting effects and centroiding performances of the methods with Gaussian and super-Gaussian approximations. The fitting results show that the super-Gaussian obtains more reasonable fitting effects. The super-Gauss orders are only slightly larger than 2 means that the CoC has a similar profile with Airy disk in certain conditions. The results of centroids computation demonstrate that when the signal to noise ratio (SNR) is falling, the centroid computed by super-Gaussian method has a less shift and the shift grows at a slower pace. It implies that the super-Gaussian has a better anti-noise capability in centroid computation.

Indefinite integrals of products of special functions

ABSTRACTA method is given for deriving indefinite integrals involving squares and other products of functions which are solutions of second-order linear differential equations. Several variations of the method are presented, which applies directly to functions which obey homogeneous differential equations. However, functions which obey inhomogeneous equations can be incorporated into the products and examples are given of integrals involving products of Bessel functions combined with Lommel, Anger and Weber functions. Many new integrals are derived for a selection of special functions, including Bessel functions, associated Legendre functions, and elliptic integrals. A number of integrals of products of Gauss hypergeometric functions are also presented, which seem to be the first integrals of this type. All results presented have been numerically checked with Mathematica.