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A useful family of integral operators and special functions plays a crucial role on the study of mathematical and applied sciences. The purpose of the present paper is to give sufficient conditions for the families of integral operators, which involve the normalized forms of the generalized Lommel functions of the first kind to be univalent in the open unit disk. Furthermore, we determine the order of the convexity of the families of integral operators. In order to prove main results, we use differential inequalities for the Lommel functions of the first kind together with some known properties in connection with the integral operators which we have considered in this paper. We also indicate the connections of the results presented here with those in several earlier works on the subject of our investigation. Moreover, some graphical illustrations are provided in support of the results proved in this paper.

In this paper, we study some properties of the locus of zeros of the Lommel function of two variables Uν(ω,z). Furthermore, we find the regions on which the curves of zeros of Uν(ω,z) lie, when ν i...

In this paper, we present two new unified integral formulas involving a generalized V-function. Some interesting special cases of the main results are also considered in the form of corollaries. Due to the general nature of the V-function, several results involving different special functions such as the exponential function, the Mittag-Leffler function, the Lommel function, the Struve function, the Wright generalized Bessel function, the Bessel function and the generalized hypergeometric function are obtained by specializing the parameters in the presented formulas. More results are also discussed in detail.

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.

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., Comment: 13 pages

Sufficient conditions are obtained on the parameters of Lommel function of the first kind, generalized Struve function of the first kind, and the confluent hypergeometric function under which these special functions become exponential convex and exponential starlike in the open unit disk. The method of differential subordination is employed in proving the results. Few examples are also provided to illustrate the results obtained.

We obtain several new results for Neumann series of Bessel functions as well as for its various special cases. The generalization of some well-known results for these kind of series, such as the Graf's addition theorem, are also established.

A method given recently for deriving indefinite integrals of special functions which satisfy homogeneous second-order linear differential equations has been extended to include functions which obey inhomogeneous equations. The extended method has been applied to derive indefinite integrals for the Lommel functions, which obey an inhomogeneous Bessel equation. The method allows integrals to be derived for the inhomogeneous equation in a manner which closely parallels the homogeneous case, and a number of new Lommel integrals are derived which have well-known Bessel analogues. Results will be presented separately for other special functions which obey inhomogeneous second-order linear equations.

In this paper, an efficient Clenshaw---Curtis---Filon---type method is presented for approximation of the highly oscillatory finite Hankel transform ?01f(x)H?(1)(?x)dx${{\int }_{0}^{1}}f(x)H_{\nu }^{(1)}(\omega x)dx$, which arises in acoustic and electromagnetic scattering problems. This method is based on Fast Fourier Transform (FFT) and fast computation of the modified moments by using Meijer G---function and Lommel function. Moreover, the method shares the property that the higher the frequency ?, the higher the precision. In particular, for each fixed ? the method is uniformly convergent as N tends to infinity, where (N+1) is the number of Clenshaw---Curtis points ci=(1+ cos(i?/N))/2,i=0,? ,N. Also, the corresponding error bound in inverse powers of ? for this method for the integral is presented. The efficiency and accuracy of the proposed method are illustrated by numerical examples.

In this work we present some geometric properties (like star-likeness and convexity of order α and also close-to-convexity of order(1+α)/2) for normalized of Lommel functions of the first kind. In orderto prove our main results, we use the technique of differential subordina-tions and some inequalities. Furthermore, we present some applicationsof convexity involving Lommel functions associated with the Hardy spaceof analytic functions, i.e., we obtain conditions for the function h µ,v (z)to belong to the Hardy space H p . 1. IntroductionLet H be the set of all analytic functions on the open unit disk U ={z: |z

The sharp version of an inequality is deduced. This inequality has been applied in several papers for the study of Bessel functions. The basic tool of this work is the development of functions in function series.

We consider a new family of nonparaxial diffraction-free laser beams with their complex amplitude being proportional to the n-th order Lommel function of two variables. Therefore, these beams are called Lommel beams (L-beams). We obtained explicit analytical expressions for the angular spectrum of plane waves and for the orbital angular momentum of the L-beams. Transverse intensity of the L-beams has a reflective symmetry with respect to both Cartesian coordinate axes. Since transverse intensity distribution of L-beams does not change upon propagation, L-beams are modes of free space (L-modes). Functions of complex amplitudes of even (n = 2p) and odd (n = 2p + 1) L-modes are mutually orthogonal. For certain parameter, L-modes become traditional Bessel modes.

The main object of this paper is to give sufficient conditions for certain families of integral operators which are defined by means of the normalized form of the generalized Bessel functions to be convex of given order in the open unit disk. In particular cases, we find the corresponding simple conditions for integral operators involving the Bessel function, the modified Bessel function and the spherical Bessel function.

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 \(s_{\mu ,1/2}(x)\), \(-3/2 0\).

Lommel's function s μ,ν(z) is a particular solution of the differential equation z 2y″ + zy′ + (z 2 - ν 2)y = z μ+1. Here we present estimates and monotonicity properties of the positive zeros of s μ-1/2,1/2(z) when μ ∈ (0, 1). The positivity of a closely related integral is also considered. © 2012 American Mathematical Society. 140 9 3091 3100 Cited By :8

In this paper, we give a set of sufficient conditions for the normalized form of the generalized Bessel function to be univalent in the open unit disk, and further we obtain certain inequalities containing normalized Bessel functions.

We will solve the inhomogeneous Bessel's differential equationx2y″(x)+xy′(x)+(x2-ν2)y(x)=∑m=0∞amxm, whereνis a positive nonintegral number and apply this result for approximating analytic functions of a special type by the Bessel functions of fractional order.

The aim of the present work is to develop a pair of Lommel matrix functions of two variables suggested by the Bessel matrix functions and some of their properties are studied to be special cases of our results.

We construct ladder operators for spherical Bessel functions of arbitrary order. Our ladder operators act independently on two parameters, one of which is the order of the spherical Bessel function, while the other parameter is a multiplicative factor in the spherical Bessel function’s argument.

In this paper, we extend some known elementary trigonometric inequalities, and their hyperbolic analogues to Bessel and modified Bessel functions of the first kind. In order to prove our main results, we present some monotonicity and convexity properties of some functions involving Bessel and modified Bessel functions of the first kind. We also deduce some Turan and Lazarevic-type inequalities for the confluent hypergeometric functions.

In the study of photon-state transitions, we found a natural extension of the first kind of Bessel functions that extends both the range and domain of the Bessel functions from the real number field to the complex number field. We term the extended Bessel functions as phased Bessel functions. This extension is completely different from the traditional “analytical extension”. The new complex Bessel functions satisfy addition, subtraction, and recurrence theorems in a complex range and a complex domain. These theorems provide short cuts in calculations. The single-phased Bessel functions are generalized to multiple-phased Bessel functions to describe various photon-state transitions.PACS Nos.: 02.30.Gp, 32.80.Rm, 42.50.Hz

The main aim of this article is to establish certain closed expressions for the Schlomilch series which members contain some Lommel functions of the first kind.

Asymptotic expansions for an incomplete Bessel function of large argument are derived when the parametric point (a) is well away from any saddle point, (b) coincides with a saddle point and (c) is in the neighbourhood of a saddle point.

In this article, we have studied the compound Bessel ultra-hyperbolic equation of the form ∑ r = 0 m C r □ B r u ( x ) = f ( x ) , where □ B r is the Bessel ultra-hyperbolic type operator iterated r-times, f is a given generalized function, u is an unknown function, x ∈ R n + and Cr is a constant. In this work we study the weak solution u(x) of above the equation which is of the form Bessel Diamond operator and moreover, such a solution is unique.